I played this position many times with white and f3 is really good move here.
It is not only about what white wants, but also about what black wants. You have more space and so you can choose easier from more plans. You can play for Rb1 b4, you can play for g4, you can play for f4 and e5. Black has not that much space and can't choose from many plans.
Now look at the position. Who will be happy to see f4 and potentially e5? Probably you in the beginning, but then also Qd8, Re8, Nf6 and Nc7. That is quite a body count. They don't have much work now but will have plenty of job after you weaken the centre. What about your pieces? Both knights will lose part of its mobility as they will have to protect pawn on e4. Bishop on c1 will hate you. Maybe you will help Rf1, while Qd1 and Be2 will see also slight improvement of their chances, but simple adding pros and cons of f4 action compared to f3 should lead to clear NO!. You won't loose the e4 pawn just by playing f4 or anything like that, but by this overextending on part of board where you really are not stronger, you achieve nothing positive at all.
The plan is good only when it helps you more than your opponent!
The f3 alternative with killing opponent's bishop immediately on c4 (f3 Ba6 Nc4 Bxc4 Bxc4 is the main line and for good reason, as black can't afford you to play b3 and retake on c4 with pawn!) saves you material advantage of bishop vs. knight, doesn't commit you to anything and then you can calmly go either for Rb1 b4 or some other stuff depending on black answer.