Learn Spanish (Latin America): El payaso tropezó y la chica se echó a reír. - The clown tripped and the girl burst out laughing.

You are right that echar means "to throw," but the reflexive phrase echarse a followed by an infinitive is a special structure. It means to suddenly start doing something. You can think of it literally as "she threw herself into laughing," which gives us the sudden, explosive feeling of "burst out."

Could I just say la chica empezó a reír?

Grammatically, yes! However, empezó a reír just states a fact: "she started to laugh." Using se echó a emphasizes that the laughter was sudden, spontaneous, and maybe a bit uncontrollable.

Can I use echarse a with any verb to mean "suddenly start"?

It is mostly restricted to a few specific verbs involving sudden physical or emotional reactions. You will almost always see it with reír (to laugh), llorar (to cry), correr (to run), or volar (to fly). For most other actions, you would stick to empezar a or ponerse a.

Why are both tropezó and echó in the preterite tense, rather than the imperfect?

The preterite is used here because both are sudden, completed actions that happen one after the other, moving the sequence of events forward. If you used the imperfect for the first verb (tropezaba), it would mean the clown was in the middle of tripping while something else happened.