To threaten the d4-Pawn.
Why Bg4 here?

Yes.
The Bg4 move pins White's f3-Knight against the Queen. What is the Knight supposed to be protecting? THE D4 PAWN.
What White piece protects that attacked Knight on f3? The White Queen. Didn't she already HAVE a job... PROTECTING THE D4 PAWN,
Black's Bg4 move threatens to knock out TWO of the defenders of White's d4 square... the Knight (by exchanging it off) and the Queen (by decoying it away).

Yes, but if you see my question is on Bh5 instead Bxf3. So, the bishop isn't doing what you are saying.

That's because White answered with Be3 supporting the threatened d4 square.
Black's Bg4 forced White's Be3 in reply... where it is not ideally placed, since it interrupts the Rook's view of the e4-square.
The maneuvering in this early stage is over control of d4. THEN the play shifts to the Queen's side with N-a5-c4 and c7-c5.
Are you sure you should be studying these opening lines if you don't understand the point of the moves? What benefit do you expect to gain from that?

"Are you sure you should be studying these opening lines if you don't understand the point of the moves?"
1) I think I SHOULD be studying these lines so I can understand the point of the moves. If I could understand the point then I would understand why I should play Bg4, then I would play it. Doesn't that make sense? Why neglect studying Bg4 and avoid playing it game after game and lose in the same line?
2) As mentioned in 1, this is not a one time occurrence. I have experienced the same issue with the London and maybe for other reasons in the QGD multiple times.
I still don't understand why black wants to cage in the bishop on g6. I'll try Bg4-Bh5-Bg6 with Na5 as mentioned.
"What did you mean by:
9...Na5 $6
{(0.19 â 0.94)"
Apparently that was the computer evaluation. It is not letting me paste the games in a diagram which I would LOVE to see so I can understand this more. You are just seeing the remnants of when I copied and pasted because this is the only way I know how to share the game.

The Bishop on g6 exerts pressure on the e4-Pawn. Since Black is playing exd4 (opening the e-file), his Rook can later come to e8 to reinforce that pressure. His c7-Pawn will advance to c5 to pressure the d4-Pawn, and after N(f6)-d7 his e7-Bishop can play to f6, both reinforcing the attack on d4 and also clearing the e-file for the e8-Rook to cooperate with the g6-Bishop against the e4 square.
The theme is converging pressure on White's e4/d4 Pawn duo.
you play 10..Bh5 to stop h3 from having purpose. H3 helps him by giving him luft, so that is why bh5 is good.
It is not like na5 is a terrible move; it is a good move and is not damaging at all. It is just that bg4 was minimally better.

The Bg4 move doesn't have the same bite if your Knight has already moved away and there is no threat to d4. Note that White isn't forced to play Be3 (obstructing his Rook) in your line.
White's center Pawn duo is under considerably less pressure in this line.
Also, your b6-Knight doesn't have much to show for the four moves it's made.

"The Bg4 move doesn't have the same bite if your Knight has already moved away"
But it is played in this game before any LSB move.

Yes, I am trying to understand this. Why can black play Na5 in the Chigorin variation but not in the Yates? I understand you are looking at it like identifying your pet dog vs. someone else's but I don't own a dog so there is no facial recognition yet. How do you know puppy Yates doesn't like to bite on the Na5 toy?
This is a common occurrence, and maybe it should go in the openings section. However, it is more to do with a specific move Bg4. I have seen this come up before in not only the Ruy Lopez but also London games. In QGD, it might also happen (for different or same reasons I am not clear on).
I'll try posting the game, if not here is the .pgn. It looks like the LSB just gets caged in on g6 regardless. I tried playing Na5 which the engine says is good, but I just should have done it later after Bg4. I don't understand why though. Any help on this would be appreciated. Thanks in advance.
PGN
1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Ba4 b5 5.Bb3 Nf6 6.O-O Be7 7.Re1 O-O 8.c3 d6 9.d4
{C91 Ruy Lopez: Closed, Yates Variation}
9...Na5 $6
{(0.19 â 0.94) Inaccuracy. Bg4 was best.}
( 9...Bg4 10.Be3 Bh5 11.h3 exd4 12.cxd4 Na5 13.Bc2 c5 14.g4 Bg6 15.Nbd2 Nd7 16.Rc1 )
10.Bc2 1-0
I'll try to paste it into a board here.