The goal of the Jerome Gambit (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+) is to attack the enemy King.
The following game is a good example.
Witzelsucht - Dencoln
3 2 blitz, 2023 Autumn Marathon, 2023
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+
4...Kxf7 5.Nxe5+ Nxe5 6.Qh5+ Ng6
7.Qd5+ Ke8 8.Qxc5 d6 9.Qe3 Nf6 10.Nc3 Qe7 11.O-O Kf7
White's King has castled. Black's King is halfway there.
Black has the standard piece-for-two-pawns material advantage that the defender has in the Jerome Gambit; but he has to be ready for the advance of the "Jerome pawns" and the following attack.
12.f4 Re8 13.f5 Ne5 14.d4 Nc6 15.Bd2
15...Bxf5
Black makes an interesting mistake, the kind that can happen in blitz games. He sees that White's e-pawn is pinned (16.exf5 Qxe3+ 17.Bxe3 Rxe3) but overlooks the protection of the f-pawn by the Rook.
16.Rxf5 g6 17.Rxf6+
“When you come to a fork in the road, take it” said baseball player Yogi Berra.
17...Kxf6 18.Rf1+ Kg7 19.Qh6+ Kg8 20.Nd5
White's pieces continue to swarm.
20...Qe6 21.Nf6+ Kh8 22.Qxh7 checkmate