They've already shot down multiple ballistic missiles / drones targeting Israel and is a counter balance so Hezbollah and Syrian militias don't get involved which will create a situation where a much bigger US response will be needed. Deterrence usually reduces conflict more than instigates it.

They are also protecting commercial civilian ships being targeted. China has warships there too and they didn't bother.

Houthi are going after Israeli stuff anyway and only targeting the US because they completely neuter their capabilities. If the US isn't there missiles will still be flying over the Red Sea and civilian ships being hijacked.