To Start Running
To Rush Out, To Run Out
intransitive verb, godan verb
You know 駆ける is "to gallop" and 出す is "to let out." Well, in order to gallop off and let yourself out of somewhere, you're going to need to start running first.
Fundamentally, 駆け出す just describes the act of beginning to run, but it's often used to mean to rush out of somewhere. For example, schoolchildren might 駆け出す out of their classroom and make a beeline for the playground when they hear the bell ring.
The readings come from 駆ける and 出す.
彼に質問を尋ねた時、駆け出しただけだ。
When I asked him a question he just ran off.