Josh says f5 ... Is Josh wrong about f5?
Yes, although black's position is terrible in any case.
If we look at the initial position and try and assess a number of things are immediately obvious:
- White has a lot more space
- All the pieces and all the pawns are still on the board
We can also say:
- White has completed development
- Black still has several pieces on the back rank and unless black can open the h file the rook on h8 is doomed to a miserable life in the near future
So, white has a very big advantage, probably close to winning.
The first two points (space advantage plus all 32 pieces still on the board) mean that it is much easier for white to move the pieces around and very difficult for black. Black's pieces get in the way of each other and there just isn't enough space in the black position for black's pieces. For black, knights are also better than bishops at the moment.
So, what can black do? Well piece exchanges would be very nice. They would create more space in black's position and make it less cramped.
Exchanging bishops would be particularly nice because, as luck would have it, the g7 bishop is occupying a square which would be great for the knight stuck on e8 and the d7 bishop is on a square which would be better for the b8 knight.
With all that in mind the pluses of h6 stand out. Bf8, the only move to not immediately lose material, firmly and further entombs the rook on h8, keeps all the pieces on the board, puts yet another piece on the back rank and moves the bishop off what is the best available diagonal for it to a dead-end square with no prospects. It becomes not much more than a "tall pawn".
What about f5? That allows Be5+, either immediately, or after a zwischenzug like gxh5 or Bc8 (clearing d7 for the knight). Let's look at what would likely happen after an immediate Be5+ and evaluate the resulting position:
[fen "rn1qn1kr/p2bppbp/1p1p2p1/2pP2PP/4PP2/2P1BN1B/PPNQ3K/4RR2 w - - 0 1"]
1.f5 Be5+ 2.Bf4 Bxf4 3. Qxf4 Ng7 4. h6 Nh5 5. Qh4 Bb5 6. Rf2 Nd7
What you have is chalk-and-cheese. Suddenly black's pieces can breathe. Black is still much worse. The rook on h8 still has no prospects but two more pieces are off the back rank and a bishop which was going to be useless has been exchanged off.
Bxh3 costs Black 2 pawns
Nonsense! I think you are making the mistake of confusing the beginner's "point count" system of 1 point for a pawn, 3 points for a bishop with the real world. There is no way the black bishop on f8 is worth anything like 3 points. Similarly the white pawn on h6 is worth a lot more than 1 point.
Is Bxh3 really the best or at least strictly better than Bf8?
Again let's try looking at a line where black follows Bxh3 with the Nxg7 immediately (the line where black plays Bxf1 is also possible although that puts black one rook exchange closer to a lost endgame) to see what is really involved:
[fen "rn1qn1kr/p2bppbp/1p1p2p1/2pP2PP/4PP2/2P1BN1B/PPNQ3K/4RR2 w - - 0 1"]
1. h6 Bxh3 (1...Bf8) 2. hxg7 Nxg7 3. Kxh3
After 1...Bf8 white has a really obvious plan to take advantage of the weak long diagonal and the monster pawn on h6 - c4, b2, line up the bishop and queen on the diagonal, mate. Alternatively, just leave black effectively a rook down (the piece on h8 is useless) and just break through in the center and mate black that way.
Going back to "beginner's point count" that rook is worth a lot closer to 1 point than 5. Once you get past beginner level you have to value your pieces according to what they can do now and in the near future.
After the sequence ending 3. Kxh3 those threats have disappeared. Black has an outpost for the knight on h5. Black might entertain hopes of playing h6 to open the h file and give the rook on h8 a purpose. Granted the position is still lost but at least there are prospects of counterplay and putting up a fight.