Why give up the bishop pair so early on in this position?

I don't see the advantage. Stockfish evaluates it about a whole point difference. Is this just Stockfish being "fishy" or is there rhyme and reason to play Bxf3 in this position? Why give up the bishop pair and allow for white to develop the queen at the same time?

The bishop pair is only of value if it can be used effectively. If the bishop goes to d7 it is not a happy piece, especially after e6. It does nothing here and also raises the question why play Bg4 in the first place instead of just Bd7? Bh5 is no better as white can just go g4, he hasn't castled yet so is not exposing his king. So it makes most sense to remove the Nf3 as this is a decent knight, helping white to dominate the e5 square. 

Ok, but the same thing happened in this game. Bxf3 was still played, and it is much earlier in the game. How can you say the bishop has no future here?

In the second example it looks a little early to my eyes to play Bg4, but ok who am I to question Fabiano Caruana. Anyway, if you are going to play Bg4 and white goes h3 you are almost obliged to then play Bxf3. Again if you retreat back along the same diagonal you have just wasted time, you could have gone directly to f5 or d7 without giving white the useful h3 (although those 2 moves make little sense to me). If you go Bh5 white can again go g4 as he has not committed his king to that side yet, then after Bg6 white can play Ne5 and exchange the bishop anyway - with additional space advantage than the Bxf3 Qxf3 line. 

Because your bishop is garbadge after Bd7 and e6. Also your position is cramped, so trades will ease the position.

In the Oparin vs Caruana game, what central formation did Black adopt after the Bishop-for-Knight exchange? If he played e6 and c6, then the exchange on f3 is perfectly logical since the light-squared Bishop has no prospects in that formation anyway.

Compare the Two Knights variation of the Caro-Kann, where Bg4 and Bxf3 is considered to be Black's most reliable system:

Here's the game you referenced:

Notice Black played 4...e5

Why? I don't know

Trading the bishop to then open up the center doesn't make much sense to me.

Notice the difference, in your game...

Nc6 was played early, blocking the c-pawn

Here, White has no issues playing c4 and maybe a later c5

My previous comment (Nc6) was if this were a Queen's Bishop game.

It could be a Chigorin also.

This is the whole idea of Bg4, taking the Nf3 to control centre, thus playing e5 and having a good centre.

Not so simple though, because the centre is then attacked in french fashion.

Sure e6 would be another approach, more solid, but playing for e5 has some points.

Like this well known line against 1.f4 :

Black just wants to play e5, so Bg4-Bxf3 is logical.

"This is the whole idea of Bg4, taking the Nf3 to control centre, thus playing e5 and having a good centre."

Hmm, ok, so the point is not the bishop vs. knight. It is to attack in the center.

Notice how all of black's pawns were on light squares... if black kept his bishop he would have been cramped.