Learn Polish: Dlaczego pan nagle się zaśmiał? - Why did you burst out laughing suddenly, sir?

It adds an inchoative aspect, meaning it marks the sudden start of an action. While śmiać się means 'to laugh' (an ongoing action), zaśmiać się means 'to burst out laughing'. The adverb nagle (suddenly) perfectly matches this sudden onset.

Could we just use the imperfective verb śmiał się since the word nagle already tells us it happened suddenly?

No, that would sound unnatural. The word nagle describes a discrete, sudden event, which requires a perfective verb. The za- prefix is necessary here to turn the ongoing action of laughing into a sudden, perfective event.

Why is the reflexive pronoun się placed before the verb instead of after it?

In Polish, the word się strongly avoids being the very last word in a sentence. Placing it before the verb (się zaśmiał) sounds much more natural, as it leans on the preceding adverb nagle.

The English translation says 'did you', so why is the verb zaśmiał conjugated in the third-person 'he' form?

Formal address in Polish using pan (for men) or pani (for women) always requires third-person verbs. You are literally asking 'Why did the gentleman burst out laughing?', which is the correct way to say 'Why did you...' in a formal context.