[FEN ""] 1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Bxc6 dxc6 5. Nc3 f6
Theory recommends 5. ... f6. Why? What is the specific reason behind this move? How this move helps Black? Other than 5. ... f6, are there any other good moves available for Black?
Not only does 5...f6 protect the e5-pawn, but it also resists against White pushing his own pawn to e5 (if Black's e5-pawn gets exchanged off).
Consider the main line of 5...Bd6: 6.d4 exd4 7.Qxd4. Now White's threatening e5, along with Qxg7. Black's only move to maintain the balance is thus 7...f6. So he ends up having to play the move anyway, but perhaps under less favourable circumstances.
Comparatively: 5...f6 6.d4 exd4 7.Qxd4 Qxd4 8.Nxd4 sees Black exchanging off the queens, making it much safer for his king (which is still a fair number of moves away from castling).
All this being said, 5...Bd6 is completely playable, along with 5...Bc5 and 5...Bg4. But 5...f6 is simply the most solid and safest way to reinforce the e5-pawn.
Funny side note, on live book someone let Stockfish somehow think to depth 53, and it gives 5...f6 as the best move with an evaluation of -0.18.
White is threatening to take the pawn on e5. It couldn't take it with 5.Nxe5 because, since the pawn on d4 is undefended, Black can immediately get it back with ...Qd4 6.Nf3 Qxe4+. Or Black could also consider ...Qg5 followed by ...Qxg2. With 5.Nc3, the e4 pawn is defended, making the threat of taking on e5 more serious. If you look at games between highly rated players, 5...f6 is indeed the most popular move, but there are other alternatives that are also fairly frequent, such as ...Qd6 and ...Bd6. I'm pasting below the table from the lichess opening book. If you look at all the popular moves (say those with more than 10 games), they all defend the pawn one way or another (in the case of 5...Bc5 it is a bit tactical).
That said, I don't know why defending with ...f6 is the most popular. It is simple and solid, I suppose (maybe they prefer to defend a pawn with a pawn to give more freedom to the pieces?). If you go by results, protecting the pawn with the queen or bishop (the second and third choices above) are also fairly common moves and actually have a higher frequency of wins for Black (but also for White!).
I think it has more to do with the surviving bishop for white. Since it covers black squares then this move indeed closes the board for that bishop.