From what I’ve understood, a player can receive feats from the DM at opportune moments when it makes most sense.

So if I play variant human and choose a feat, instead of normal human with +1 bonus to all my ability scores, I could have just gone for the exact same flavour with normal human and received the feat for free from the DM?

Likewise if I trade an ASI for a feat, I could have improved my scores instead and received the feat for free again?

Please explain that I’m wrong. This doesn’t make sense to me, and seems unfair.

When you level up and get an Ability Score Increase, there is an optional rule that you can get a Feat instead of improving Ability Scores.  That is where the balance is.
For a human... you can go with the standard human and get 6 +1 Ability Score bonuses or you can choose the Variant version and get a Feat and 2 +1 Ability Scores.
Those are both RAW and RAI... so no issue.
It is very rare that a DM will hand out feats for free at opportune moments.  There are some of us that do (or as part of downtime training), but it is not RAI or RAW.  Usually when this kind of scenario happens there should be balance with the party... but as it isn't a written rule, it is up to individual DMs.

The DM can award feats at moments where they believe it would enhance the story.

That doesn't mean the DM will do that thing, or if they do that they will do it often.  

As well, most of the feats players want are things the DM would not award as boons.  DM-awarded boons are usually less powerful, lesser-used feats like Inspiring Leader given to characters who've naturally demonstrated the abilities the feat was built to emulate.  If you want Big Damage Combat Time feats like Polearm Master, Crossbow Expert, or Sentinel, you almost always have to take those as a level-up option.

Some DMs will allow a player to work towards a feat as part of their character's ongoing story, but again - this is usually something done for one of the lesser-seen feats, and it usually takes quite a few sessions before such a plan pays off.

You should never assume your DM will give you free feats unless your DM specifically says they're going to do free feats throughout the story.  And even then, getting a feat a few levels early may be worth it, depending on how important to your character that feat is.

This entirely depends on the DM. Think of it a bit like finding a Magic Item... many Magic Items will give boosts or abilities equivalent or better than what you can get from feats. A DM may choose to grant a feat to a player outside of level progression for story or gameplay reasons. A famous version I can think of is from Critical Role, where one player completed an important quest for their character and as a result was given the Inspiring Leader feat... which ultimately wasn't THAT useful, since the character had low charisma, but it was a nice little bonus for the player.

Feats are optional rules to begin with... some DM's won't use them at all, and some may restrict specific feats (I know that Lucky is banned at quite a few tables). I think the reason it gets mentioned in one of the books that feats can be awarded outside of level progression is just to help inspire DMs... nothing is stopping a DM from giving any feat however and whenever they want, but this lets them know that it's not out of the ordinary and is a safe thing to do in their games.