To Spit Out
To Throw Something Up
transitive verb, godan verb
You know 吐く is "to throw up," and 出す is "to take out" or "to let out." So you're not just throwing up here, you're letting it out into the world on purpose, which makes this to spit out or to throw something up.
吐き出す can be literal, as in 水を吐き出す (to spit out water), or figurative, as in 気持ちを吐き出す (to let out all your feelings).
The reading comes from 吐く and 出す.
男は舐めていた飴を吐き出しました。
The man spit out the candy he'd been sucking.