How have others gone about doing this in their world building or even their character creation? In my group we have a mix of races from different settings including Theros, Eberron and Ravinca. Specifically I am creating a Leonin character and the other characters being created are a Satyr, a Warforged, a Khalastar, a Loxodon and a Vedalken. The DM is open to having any player race option available, but doesn't want to create a whole new world setting. Do we just play as if each place is somehow on the same plane of existence? 

I guess you can just add them in. Leonins live in the mountainous area (making this stuff up, just as an example), warforged hang out with gnomes cause they made them and satyrs, loxodons, minotaur and centaurs live in grassland areas etc. You can change it to your world, without making a while new world. I added in elves, dwarves etc to my Runeterra campaign, even though that world doesn't officially have them. 

The beauty of D&D is the rules are meant to be broken. As BramblefootDruid said, just say they're all rare travelers from an area that's rarely visited in whatever setting you're using. Or if you really want to stay in "canon," just run Curse of Strahd and say they ended up in Barovia from their respective settings. 😜

Quote from CaptGladMane >>

How have others gone about doing this in their world building or even their character creation? In my group we have a mix of races from different settings including Theros, Eberron and Ravinca. Specifically I am creating a Leonin character and the other characters being created are a Satyr, a Warforged, a Khalastar, a Loxodon and a Vedalken. The DM is open to having any player race option available, but doesn't want to create a whole new world setting. Do we just play as if each place is somehow on the same plane of existence? 

Brambledruid is on the money.

You can also say the other characters foreign to the setting are Planeswalkers who ended up there by accident.

In my Tomb of annihilation game I got a Minotaur paladin from Ravnica who crashed thete due to magical shenanigans and he is searching for a way home.

Like Sliders!

Also, all of those come from other material planes. As I mentioned on the similar thread about warforged, travel between the material planes is 100% canon. All you need is a spaceship. Literally. A spelljammer, which is a ship that flies through space. So there's that.

Warforged, Kalashtar, and possibly Simic Hybrid are the only ones I feel like you really have to explain away, just since they're baked into the lore of their respective lore. Even then, mad gnome artificers abound across the multiverse, and where they go, creatures like Warforged and Simic Hybrids (minus the "Simic" part of you're not playing in Ravnica) follow, so they don't require much explanation. 

In my homebrew world there's a gnome nation that engineered a small army of warforged to fight invading gnolls, and that army slowly gained sentience and rebelled and both sides petitioned an allied king to arbitrate their conflict when the fighting came to a standstill, and the king ruled in the WF's favor and boom, now there's a race of warforged in the world. 

Quote from CharlesThePlant >>

Warforged, Kalashtar, and possibly Simic Hybrid are the only ones I feel like you really have to explain away, just since they're baked into the lore of their respective lore. Even then, mad gnome artificers abound across the multiverse, and where they go, creatures like Warforged and Simic Hybrids (minus the "Simic" part of you're not playing in Ravnica) follow, so they don't require much explanation. 

In my homebrew world there's a gnome nation that engineered a small army of warforged to fight invading gnolls, and that army slowly gained sentience and rebelled and both sides petitioned an allied king to arbitrate their conflict when the fighting came to a standstill, and the king ruled in the WF's favor and boom, now there's a race of warforged in the world. 

In my ToA I introduced a rival team to my PC's, have a bit of an adventure film kinda thing, and three of them were from Lantan. Among them a Warforged.
Since Lantan is MILEs ahead technology wise compared to the rest of Faerun, I thought Warforged would make sense if they came from there. 

And any number of creatures can be one-off results of magical experiments or curses or whatever.

Adding on to what Gorvar stated earlier, Planar travel is also possible through Sigil, the City of Doors (Planescape) and that is a really fun setting.

Most races in any given D&D world are presented in such a way that they don't need a particularly complicated and involved backstory about why they exist.

Yes, there are usually proper origins for a race in a setting if you read into the books enough, but they basically never come up during the life of a campaign.

Maybe the only one where it is really important to know why they are about are Warforged, as they have an origin within living memory and came out because of a very specific event and if that event didn't happen in your world-- you need a totally other origin for them as to why people went about building a bunch of them and then disowned them.

But in terms of any naturally-breeding races go? Pretty much you can just say that they are just in the world, as far as history books record-- they have always been about in the world, and just no one is surprised to see one at this point.

Don't get hung up on the fact that the race isn't mentioned in the official handbook of the setting because the very second you sit down and start playing, you are automatically playing some variant of the setting and not the official canon version anyway. There being a new race flitting about doesn't disrupt things any more than if you just put a monster, other than like a construct or something else that could be a one-off, in there that hadn't been described by the setting before. You don't fret over how the monster fits into the ecological niche of the world and what they have been eating and how they get together with their mates in order to breed the next generation and how many of them the ecosystem can support and if your killing the monster somehow disrupts the ecological balance of the ecosystem....

No, you just encounter the monster and then you kill it. This is especially true if you are playing a classic style dungeon crawl where you just open a door and-- boom-- there is a monster and no thought has been given as to why the monster is in the room, how it got there, how long it has been there, what it has been eating, etc.

The same goes with races.

I like to create some location in the world that explains the race rather than saying they came from another plane. This lets you tie in NPCs the PC will care about and tie their backstory into whatever greater story exists. For example, I am currently running Tyranny of Dragons for my group and one player is playing a Loxodon Barbarian. I set it up that the Loxodon are from the east, in a desert near Thay. The Thayan's have been subjugating the Loxodon for centuries, taxing them and raising their dead to fuel Szass Tam's Legion of Bones. This will hopefully create a nice tension when it comes to that PC dealing with the Red Wizards Helping the Cult of Dragons. He will share their goal of overthrowing Szass Tam, but will probably disagree with their methods.

Forgotten Realms, or at least the 5e SCAG specifically has a passage discussing how whole continents populated by intelligent people are sometimes come across on the trackless sea, and then those continents are never seen again.  Having a PC come from a place like that, and they got on a sailing vessel to explore, not realizing their "home" wouldn't be waiting for them is a common way to introduce (let's not use interweave) a non traditional PC race into the game (and give precedent for encountering more here and there.

This is why I push back hard against those who ignore the DMG's world design guidance (summed up in two words "start small") and instead feel the need premap an entire world with. complete demographic info etc.  This is a false problem if you pay attention to the DMG's guidelines.  The PCs come together from near and far and cotend likely with some local challenges in the early game.  DM will let you know if your species in a major player or presence in your present environment, and if not, they're some sort of "far traveller type."  Does it matter what their place of origin looks like or where it is on a map?  Not really, definitely not until the DMs played out their first few ideas for the campaign.  Eventually it may matter but things like "where did I come from and what's going on there now" are things that at the start of a game are on the horizon and the PCs can barely see the horizon.  

It's not the best organized sections, but if players and. DMs really paid attention to the relevant sections in the PHB and DMG, they should realize they shouldn't show up with an encylopedic understanding of their characters or their world.  Those moves limit growth.  Show up in character sheet and "built" world with a lot of loose ends.  Let the game play out by pulling at them, and see where they go as the game's played, not some articulation of the wheel of fate.

Admittedly this won't work for everyone: but this is actually a core plot point of a campaign I'm in.  Basically: the entire party are "exotic" races for the setting... because tehy literally aren't form it.  Long story short: the BBEG of teh campaign literally isekai-ed (anime-derived term meaning "taken toa  different world") from their native planes of reality.  Thus; exotic races, different gods, and entirely different cultures and even technological development levels.  The entire party are functionally "aliens" in the otherwise "normal" fantasy setting.

As others have said, you can either add the race to the setting or make them tourists/immigrants/just really lost/etc from another plane.

In the former case you can either designate a specific "homeland" for them within your setting or integrate them into one or more existing societies. All you really need is at least one place where you can find a bunch of them (unless they're scattered all over in little nomadic tribes a la gypsies or such).

If they're extraplanar in origin they could have gotten there via mechanics of Spelljammer or Planescape (portals/Sigil), they could have wandered into some portal that was either created or naturally occurring, they could have been transported by a powerful spellcaster either intentionally or accidentally. Seriously, there are all sorts of possible explanations for one way planar travel in D&D with the most basic being some variation of "a wizard did it."

Something else to bare in mind when dealing with "exotic" races; how they are treated may vary wildly dependant upon the setting.  Recall that, for most of our history, information spread VERY little; what was on the other side of the world might as well have been on another planet, for instance: when provincial Roman soldiers first saw Hannibal's elephants: that was probably as close as humans got to fighting monsters; because they had precisely no idea what those creatures were more than likely.  So; say your average D&D universe peasant meets a Genasi; they might just assume "well; that's just what a genie looks like I guess..." or "Must be some wizard's construct or something".  That doesn't even necessarily mean xenophobic responses; the average D&D universe has a lot of weird stuff running about in it, so your average innkeeper probably thinks "Eh; if it's got two legs, walks upright, isn't undead, and has coin I'll just roll with it."

When in doubt, there is the World Serpent Inn:

https://forgottenrealms.fandom.com/wiki/World_Serpent_Inn

With DM's approval teleport them in. 

Sure you could just say that all those settings are just distant places on the same world. No problem with that. 

But if the DM doesn't mind doing a bit of extra work, perhaps the whole basis of the campaign is that there have been some kind of rifts happening between planes that have been forming random or accidental gateways between planes. And maybe that's how you all met. Maybe each of you walked through a doorway on your home plane, or your home setting, and somehow got bamfed to this plane. And maybe the plot of the campaign is that you guys are investigating these planar rifts and maybe trying to find a way to repair them.

But... a problem arises when you realize that once you close the rift to your home plane, you won't be able to go home. So then each player is forced to make a choice - once the rift to their home plane is sealed, do they go home (and thus leave the campaign), or do they forsake ever going home in order to keep helping to close the other rifts?

Just my 2 c.p.

DM approved multi-verse travel. 

Yes - DM approved/authorized multiverse travel. Works whether you are a strange race or just a regular character of a player that moves a lot and brings their characters with them. 
You go to sleep camped in a foggy area and when you get up the next morning some practical joker of a deity has shifted you to a new world for some reason - good luck figuring out why!