Hi guys, first time DM here currently writing my own campaign! 

I have an NPC wizard who basically offers her services in assisting other NPCs to train in fighting. I'd like for her to offer this for the party too, and create a kind of fake reality area where players can fight low level monsters once or twice a day. There is no gold achieved from this only a small amount of XP. 

I was just hoping for some advice on how I should go about this? I want to create some kind of safety net where once any of the members of the party are downed, they are teleported back to the real world and health is replenished or something like that. 

You've already created it in your mind. Just describe it in game like you described it here. You don't need to have an actual spell mechanic set up, just narrate what you want done and poof! If you put a hard mechanic in place, it can be broken or abused by everything in the game world.

Do you want the players to actually go through the combat? I'm afraid it might feel a bit boring and like a waste of time if the enemies are really low level, hardy give any reward, and it doesn't progress the story.

Perhaps you would want to make it a downtime activity with a few dice rolls? XGtE has a pit-fighting example as a downtime mechanic. Perhaps you can adapt that.

I have two ideas for how you can implement this.

1. Perhaps during a rest, they enter a sort of 'dream sequence'. In this dream, they are pit in a small encounter with a few somewhat challenging enemies. If any player is downed, they wake up having benefited from the rest at full hit points and with all their spells.

2. Use a modified version of Animate Objects to create a couple of 'training dummies' that actually fight. When a character is at low health or is struggling, the dummies pull their punches and help the injured trainee to their feet. The dummies are more of friendly sparring partners than actual enemies.

I like the dream sequence idea since you can put enemies together in strange and unexpected combinations to challenge your players. You likely wouldn't see matchups like those in normal adventures.

I think how you set up a "Danger Room" comes down to what is the purpose in game. 

As far as its workings I agree with others that rather than think of specific RAW magics to accomplish this just presume the NPC is a high level illusionist or has a wide range of summons/conjure magic that can bring about whatever adversary AND a way to plop the baddie back wherever it was plucked from in the first place if things get too "dicey" in the arena.  [PLOT/GAME hooks: 1. Maybe these magics will turn any of your wizards on to Illusion or Conjuration magics, in my own game I have a character that's found a few elemental and other planar gems that bring force various monsters and has developed an interest in their workings to the point that they'll be able to acquire those spells when level appropriate 2. if they're not illusioned but conjured ... this place from which these beings are plucked, maybe these boundary transgressions are weakening and the NPC may "bust the tap" causing an invasion, sorta a BIG ACCIDENT instead of. BBEG).

Back to game purpose, I mean, I could see it as a sort of proving ground where characters can test out leveled up abilities and tactics in a "safe" environment, if your players are into playing out tactics to figure out optimizations and the like.  Since these combats presumably have a "safety" I'd probably offer reduced XP for the training sessions.  Pit fighting combined with some of the other training mechanics in XGtE I think might be a way if you wanted to award proficiencies and feats outside of level progression (or have a in game rationale explicit cause for those level progressions' acquisitions of proficiencies and/or feats).  Maybe "pit fight train" till the proficiency or feat is earned, then play a "match" so they can see how the new weapon or proficiency works.

As mentioned above, don't lean on this too much (unless you just want a place for the characters to fight things with little consequence) unless the party seems pretty invested from an RP standpoint ... maybe they're all fans of Spartacus: Blood and Sands, and want to get involved in the intrigue at various levels within the operation of a Ludus.  Actually, that would be a pretty cool campaign set up for a game whose real goal is to tutor new players in the 5e's martial aspects.  STEALING  

Danger room type setups can be a convenient way to teach novices how to play the game, but are rarely going to be of long term interest. In terms of mechanics, there's no canonical spells that can do exactly this, but an illusion that is only effective on willing targets, or some form of shared dream, isn't a crazy powerful custom spell.