rnb2rk1/ppq2pbp/2pppnp1/8/2PPP3/2N2N1P/PP2BPP1/R1BQK2R w KQ - 0 9
1. Bf4 Nh5 2. Bh2 e5
In the game, I played Be3; but the engine seems to like the line above. Why is Bf4 better (in a strategic way) than Be3? I mean the Bishop will be sent off to h2, and black will gain one time to play e5 and develop his queenside pieces.
Be3 is rather passive. It defends f2 and d4 but otherwise doesn't do a lot.
Bf4 is much more aggressive. It eyes up the black queen on c7 and pins the d5 pawn threatening an embarrassing c5. Black's response of putting the knight on the rim and pushing the e5 pawn is pretty much forced. Getting the pawn to e5 has taken black 2 moves whereas normally in these structures black would avhieve that in one move.
If white wants to start a quick pawn storm on the king side then the "dim" black knight on h5 gives white a quick start with g4 coming with tempo. Although white should probably worry about king safety before adopting that plan.
Since the first part of your question was already answered, I'll tell you why Bh2 is may be considered "useful".
Notice that white's c and e pawns have a strong hold on the d5 square, rendering a d6-d5 pawn push from black impossible. This in turn makes the d6 black pawn a backwards pawn (not protected by any other pawns) and notice that the Bh2 not only x-rays the queen but also puts a lot of pressure on that pawn. It also places pressure on the advanced e5 pawn which cannot move without exposing the weak d6 square to the bishop.
..e5 and ..c5 can both be strong moves if followed up consistently. Usually one of them is necessary to avoid a cramped position. In the given position Black has already played both ..e6 and .. c6, pretty much guaranteeing a lost tempo. Also it is not yet clear that the Black Queen belongs on c7.
In a Kings Indian formation Black hopes for one of six developments.
This is a very incomplete and sketchy outline of the possibilities and the choice is part personal preference and part what White has committed to. They have to be considered in conjuction with each of the three Bisop placements (e3,f4,g5 and perhaps eventually h6)
The decisive factors in this instance stem from the three weak moves (e6,c6,Qc7) that allow threats of e5 and c5 to be generated following Bf4. The strength of this move does NOT lie in its strategic ideas, as the OP suggests. In fact generally speaking, f4 is usually the weakest of the three options because it allows ..e5 with tempo. It is the tactical alignments of the Bf4 and the Qc7 , also the Be2 and the Nh5 that give the move its power. At the end of the day White will have strategic advantages, but he will only get them by using tactics that are specific to this particular position.
Sometimes the strength of a player lies in recognizing that the normal rules do not apply. Although my six maxims are the ones that every Kings Indian player must know, none of them apply to this position. The slightly unusual move c5 is the one that has to be seen, and it is enabled only by the combination of three Black errors. These weaken the square d6 and misplace the Queen.
There's a myth in these Kings Indian type positions that Black wants a pawn on e5. Actually, that's not the case; Black wants to keep the long diagonal open for their bishop if possible, and having a pawn on e5 and then making a kingside pawn storm is just the choice they are forced into when they have to give up too much to keep the diagonal open.
If you play Be3, Black can play c5, and you're pretty much forced to concede that open long diagonal to Black. On the contrary, if you can force Black to play e5, that pawn won't ever leave that square. (Even if it has a chance to move, moving it exposes a weak backward pawn on d6.) With the center blocked, you have a chance to launch an attack on Black's disorganized queenside. (In contrast, your kingside defense is reasonably well organized and you don't even have to put your king there.
