Learn French: Explique-moi simplement ce qui s'est passé.

The use of invariable adjectives as adverbs is an exception that only applies to a few specific fixed phrases, like coûter cher, sentir bon, or travailler dur. For most other verbs, including expliquer, you still need to use a standard adverb ending in -ment, which is why we use simplement here.

How do I know to use ce qui instead of ce que in this sentence?

Use ce qui when "what" is the subject of the verb that follows it, and ce que when it is the direct object. Here, the "what" is the thing that actually happened (it is the subject of s'est passé), so you must use ce qui.

Could I just say ce qui a passé to mean "what happened"?

No, because the regular verb passer usually means "to pass" (like passing time or handing someone an object). To mean "to happen" or "to take place", you must use the pronominal verb se passer. Because it is pronominal, it uses the auxiliary être in the passé composé, giving us s'est passé.

Why does moi come after the verb with a hyphen?

This sentence is an affirmative command (the imperative mood). In affirmative commands, French requires you to move the object pronoun after the verb, attach it with a hyphen, and use its stressed form (so me becomes moi).