I think white's king is in the center. Black can rip the center open with 9...cxd4 10.cxd4 e5! 11.fxe5 dxe5 and white has trouble.
Nimzo Indian

Well... apparently the idea behind 9.f4 is pushing e4-e5.
Black can now play 9...e5 himself (I don't think that taking on d4 first has any additional value) and you must be careful- you are behind in development.
And anyway- if you wanted to push the e pawn, why not 9.e5 immediately? Sure, it leaves white with a "suspect" queenside pawn structure, but this is something that white knows well when he opts for 4.a3.
So how come stockfish doesn't like White's 9th move? Yes I understand it is not a developing move, but, it is not like Black's position is highly developed either. I mean sure, he is castled, however, I saw the move f4 as a way for white to establish power in the center with pawns.