The main idea behind 4.Bxf7+ in the Jerome Gambit and related openings is to checkmate the enemy King. Tactics are the key that lead to execution, as in the following game.
Guest5673273290 - Guest2514575685
10 0 blitz, Chess.com, 2021
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6
The Two Knights Defense.
4.Bxf7+
The move seems to be growing in popularity. This game will help.
4...Kxf7 5.Nxe5+ Nxe5 6.d4 Nc6 7.e5 Qe8
According to The Database - which is somewhat incomplete on this line - this is a new move, although 7...Qe7 was seen at least as early as Philidor 1792 - guest1334, 3 0 blitz, 2014 (1-0, 20).
8.O-O Nd5
This is a reasonable move, although Stockfish 13 prefers 8...Nxe5, returning some material directly. It then further recommends 9.Re1 d6 10.dxe5 dxe5 11.Bf4 Bg4 12.Qd2 Rd8 13.Qc3 Qc6 14.Qxc6 Bxc6 15.Bxe5 Bd6 where White has a pawn and better pawn structure as (insufficient) compensation for his sacrificed piece.
9.Qf3+ Ke7
It is time for White to hunt the enemy King.
10.Bg5+ Ke6
The alternative 10...Nf6 allows White to grab material with 11.exf6+ Kf7 12.fxg7+ Kxg7 13.Bf6+ Kg8 14.Qg3+ Qg6 15.Qb3+ (with the idea of moving Black's King away from protecting his Rook) Qf7 16.Qxg7+ Kxf7 17.Bxh8 when White would be up the exchange and a couple of pawns. The Bishop will escape.
11.c4 Nxd4
Black's best was to give back a piece with 11...Nce7 12.cxd5 Nxd5. Then Stockfish 13 has an interesting Queen maneuver for White: 13.Qh3+ Kf7 14.Qh5+ g6 15.Qf3+ Kg8 16.Qxd5+ when the first player has recovered his sacrificed material.
The point of the Queen's efforts can be seen by comparing them to 13.Nc3 c6 14.Nxd5 cxd5 when White has an edge, despite being behind in material, because of the unsafe enemy King. Once the point f6 has been weakened - 14.Qh5+ g6 - then it becomes extra dangerous for Black to move his King to f6 - e.g. 15.Qf3+ Ke6 16.Nc3 when the Knight exchange is forced 16...Nxc3 17.bxc3 and the only move to prevent checkmate would then be 17...d5, although lines of attack would open after 18.exd6 e.p.
That seems like a lot to look at, but it is helpful to have a good understanding of the attacking position.
12.cxd5+ Kxe5 13.Re1+ Ne2+
Or 13...Kd6 14.Bf4+ Kc5 15.Qc3+ Kxd5 16.Qa5+ b5 17.Nc3+ Kc4 18.b3+ Kc5 19.Rxe8 Bb7 20.Na4+ Kd5 21.Re5+ Kd6 22.Qb4+ Kc6 23.Qc3+ Kd6 24.Qc5#
14.Rxe2+ Kd6 15.Qa3+
15...Kxd5 16.Nc3+ Kc6 17.Qa4+ b5 18.Qxb5+ Kd6 19.Rd1 checkmate