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Why it works
3.Nxe5! against the Damiano is correct — but only for players who know the follow-up. After 3...Qe7! (attacking the knight and preparing to meet the check) the knight must simply come home: 4.Nf3. Instead 4.Qh5+?? feels like pressing the attack, and it loses a piece by force: 4...g6! hits the queen, and after 5.Nxg6 Black has 5...Qxe4+! — a royal fork that picks up the g6-knight two moves later. White keeps checking, Black keeps taking.
Refutation
4.Nf3 (or 4.Nc4) retreats and banks the extra pawn: 4...Qxe4+ 5.Be2 regains nothing for Black, and White is simply better (engine +1.3). The whole point of meeting 3...Qe7 is modesty — the knight did its job by winning e5; asking more of the position with 4.Qh5+ is what loses it.