Learn Russian: Я хотел выбежать, но мой шнурок застрял. - I wanted to run out, but my shoelace got stuck.

You could say Я было выбежал, но мой шнурок застрял. While Я хотел выбежать simply means "I wanted to run out," using было with the past perfective verb (выбежал) emphasizes that you were on the very verge of running out—or had literally just started to—when you were abruptly interrupted.

Is the prefix вы- in выбежать what gives it the meaning of running "out"?

Yes! The prefix вы- indicates motion "out" or "out of." Added to the verb of motion бежать (to run), it creates выбежать (to run out). As a handy pronunciation rule: in perfective verbs, the вы- prefix always takes the stress, so it is pronounced вЫбежать.

Застрял (got stuck) is perfective. What is its imperfective pair, and why is the perfective required here?

The imperfective pair is застревать. We use the perfective past tense застрял here because the shoelace getting stuck was a sudden, single completed event that thwarted the main action, rather than an ongoing or repeated process.

Does the word шнурок (shoelace) have a fleeting vowel when it changes cases?