Why is Qf3 better than Nf3?

1. a general rule is to solve problems with devolopment, here your problem is your N is under attack and you threat QA8+
2. the threat of QA8 is very dangerous and is winning.

Qf3 threatens the king with Qa8 on the next move. Your oppenent could capture your knight and defend with Nb8 afterwards, but this will cost him a knight as well (and also a bishop or an exchange of queens), so he won't capture your knight.

Always keep in mind that there are four possible defences if one of your pieces is under attack:

1. Move the piece to safety, preferably somewhere where it's still dangerous. Don't retreat.
2. If the attacking piece is more valuable than yours, back it up and make it an exchange.
3. Move a cheaper piece into the line of fire and block the attack, preferably making it an exchange as well.
4. Press and unrelated attack, which forces the opponent to defend his pieces instead of attacking yours.

In your example, all four defences are available. You can move the knight to safety with Nf3, back it up with c2, block the attack with b3 or press an unrelated attack with Qf3. Qf3 is the best option, because it puts you in a great position, forces the opponent to waste a tempo defending his king, and you can still protect your knight with c2 of b3 afterwards. Retreating the knight to Nf3 is the worst option.

Welcome Ludi!

As RemcoG points out, Qf3!

This is an example of a zwischenzug or "intermediate or interpolated move". Often a surprise move or an unexpected move. Black thinks you are forced to attend to your attacked knight, but you have a surprise in store!

But, how would you come to consider such moves?

Your knight is attacked as you point out, so the first thought is to either defend it or to move it to safety.

However a useful thought pattern in chess is: I have found a good move, but is there an even better one?

This skill comes about from developing sight-of-the-board.

You can develop this skill by practise. A quick scan should allow you to notice several general things:

1. You are the exchange and two pawns up.
2. Your king is a bit exposed but your opponent does not have any effective threat against you king.
3. Your opponent's king is in a precarious position: a check delivered on a8 or c8 by either a rook or queen would be checkmate! Any threat to do this cannot be ignored so one thought is: can any of my pieces attack either of those squares? It leads you to Qf3!

This general scan of the board routinely can be done when it is your opponent's move. Then when it is your turn to move you already have a useful framework to decide on a specific move.