Why is Bxh6 always suggested as the best move? Am I not just going to lose my bishop to a pawn?

H6 is a knight and taking it doubles blacks pawns on the h file which makes them easy targets later in the game.
I mean, if a bishop has already taken a knight, it getting recaptured isn't exactly that big a deal in most cases. Here you've doubled their pawns on the side they have open to castle from while your stuff seems ready to attack on the other end.
The bishop and the knight are roughly equal in value. Here Bxh6 gxh6 will make a hole in the kingside pawn structure. It would then be risky to castle kingside. If Black wants to castle queenside, they will have to develop some pieces first, while you could castle and start your attack while the Black king is still in the center. So while you don't win any material, you will have a positional advantage. In addition, the doubled pawns are isolated (cannot be protected by pawns) and they are an easy target further in the game.
Castling is the obvious second good candidate move, the third one is h3 (preventing Ng4 or Bg4 and practically makes the knight almost useless for a while as it can only move back to g8).
The point about black's pawns structure being damaged is correct. However, in this specific case, I'd play Bg5 first here, then capture with Bxh6 after black saves his queen. All of black's responses to save the queen are weakening:
a) f6 opens the black king to attack on the h5-e8 diagonal, and further weakens the kingside pawns after Bxh6 gxh6
b) Ne7 deactivates the queen's knight and relieves the pressure on white's d4 pawn
c) Qd7 blocks in the light square bishop (this is probably best, though)
And of course, black can always blunder the queen. In any case, black has to answer the threat, so the knight stays on h6 regardless. If you mean to trade off to double up black's h-pawns, might as well use the bishop to soften black's position elsewhere before you do so.
This kind of tactic is called a zwischenzug, meaning "in-between move". Instead of doing an obviously strong capture immediately, you make a threat the opponent must answer in order to gain additional advantage after the capture takes place.
I once won a round one game at a US Open against a player 600 points lower-rated than me. After the game he said he did fairly well by keeping it materially even for more than thirty moves. I responded by saying he lost the game positionally on move six by letting me play BxN(h3) and forcing him to respond with gh and giving himself isolated doubled h pawns. At the time I made the capture I wasn't sure what moves I would end up playing to win but I simply stuck to a plan of trading most of the pieces and winning the ending due to that chronic weakness in his position.