First off, it's good that you're careful about not creating weaknesses. Noting that the e3-pawn might be vulnerable to the e8-rook is good. Many players just push pawns without considering what pawns/squares they may weaken.
In this position though, an important point is that Black's pieces are quite passive and cramped. This will make it more difficult for him to attack the e3-pawn. Also, because your opponent's pieces currently aren't good, you'd like to open up the position before they can become good.
After 11.f4, if Black plays say 11...Bf8, then Stockfish gives is 12.Qd2 g6 13.e4 Bg7 14.exf5, etc. So here the e-pawn wasn't an issue for White, as it was soon advanced. Meanwhile, after 11.f4 exf4, SF gives 12.Nxf4 Ng4 13.Qd2 Bg5 14.Ncd5, and try as he might Black is unable to win the e3-pawn. Now White will likely continue with h3 on the next move, kicking the knight off g4.
2rqr1k1/pppb2pp/2np4/3N1pb1/2P2Nn1/1P1PP1P1/PB1Q2BP/R4RK1 w - - 0 14
In this position, note how White's b2-bishop, f1-rook, and f4-knight have been improved, thanks to the f4 ...exf4 Nxf4 exchange a few moves prior. While Black's rook has also benefited from this, it is unable to work with any of its fellow pieces effectively. So White benefits more overall from these events.
The f4 push is a useful idea to remember in the e3 English. It's definitely not always good, especially if your d-pawn is no longer on d2 (as it weakens the e3-pawn). However, if you can defend the e3-pawn with many pieces, and if your opponent has difficulty attacking it, then f4 becomes more sound. The key question is whether playing f4 gives you more activity than it does your opponent, and if the problems they will have to deal with outweigh the problem of the e3-pawn for you. The position you gave is an example where the answer to this is yes.
