Why does Stockfish recommend this bishop exchange early on?

Three Options

After 7.Bd2 Black has three options:

Retreat: Black can pull the dark bishop back. Retreating to c5 doesn't work, since White can just push the b pawn forward, and force the dark bishop to retreat again, so you'd have to go back to d6 or further. The issue with this is tempo - you've giving white a free turn to develop.

Defend: You can try to use your Queen or the a pawn to defend the bishop, but these are a little awkward. In the pawn's case, White can exchange for the bishop anyway, and you are left with doubled pawns on b, and now you have to defend the forward pawn or just coincide it. The Queen defense line is more complex, and I'm not sure I'm experienced enough to do it justice. Bottom line is, your Queen is either tied down defending that bishop, since you can't get any other support there easily, or White is going to exchange for the bishop (again) and either push your Queen around or try to exchange for it.

Exchange: Or exchange for the bishop. Your example ends with White's Queen taking the dark bishop. This doesn't really develop White's Queen, so you gain tempo by following it up with a development move - maybe the knight f6 you mention in your question.

Bottom line: the bishop is probably going to be exchanged, so you might as well do it in a way that doesn't give White anything. Moving the White Queen one square forward isn't much in the way of development, so that seems a good exchange.

There isn't a sacrifice in the line you gave. I assume you man 6...Bb4+.

The bishop is a bad bishop because it's blocked by its own pawns in the center. 7.Bd2 is virtually forced trading black's bad bishop for white's good bishop. The alternatives for white just aren't very good.

It also frees up black's position because he can play d6 to develop the other bishop and he has Ne7 as an option too.

That is not a sacrifice at all. It is an exchange. You get rid of a bad bishop for his better one. It is positional play for a very minor advantage. But still an advantage. Enough small ones add up to be enough to win material and the game.

Mostly when the engine suggests a such exchange maneuverer it means that either your piece has no future, or the blocking piece has a very good future. Since you need to move the bishop somewhere to get castled, it chooses to exchange it off.

If you look for plans in the variation you given, the most natural way for White to proceed is to break in the center with f4 at some point, where you can support your center with ...d6. The bishop would find it hard to find a meaningful role in this structure.

Once white has put pawns on the central white squares of d3, c4 and e4 his white squared bishop becomes very bad, because its mobility is drastically reduced, and his dark squared bishop becomes essential for protecting the dark squares that have become weak thanks to the pawn moves.

That means that 7.Bd2 is a blunder because it allows black to exchange the dark squared bishops leaving white with a bad position. Much better would have been 7.Nd2 followed by a3 and b4 to chase the dark squared bishop away.

I wouldn't say it was a "sacrifice", it was more like an exchange of bishops.

My lichess engine was actually giving me ...Bd6 as the first move.

I don't understand why it wants me to move my bishop to b4 in the first place

Me neither. I doubt anyone here will have an idea. Stockfish doesn't tell us why. If I have to make a guess, I would say the bishop is being blocked by the d4-e5 pawn chain, so it makes sense to exchange the bad bishop off.