Learn Icelandic: Strákurinn fer af sviðinu snöktandi.

Here, snöktandi (sobbing) acts as an adverb of manner, describing how the boy leaves the stage. One of the great things about present participles (words ending in -andi) is that they are indeclinable when used like this. Whether it is one boy, a girl, or a whole crowd leaving the stage, the word stays exactly as snöktandi.

I thought fer (from fara) meant 'goes'. Why is it translated as 'leaves' here?

You are right that fara usually means 'to go'. However, when paired with the preposition af (off/from), the phrase fara af translates naturally into English as 'to leave' a specific surface or place. Literally, he goes off the stage.

Why is the word for stage, sviðinu, in the dative case here?

The case is controlled by the preposition af. In Icelandic, af (meaning 'from' or 'off') always forces the following noun into the dative case. The base word is svið (a neuter noun), its dative form is sviði, and when you add the definite article suffix, you get sviðinu.

Could I put snöktandi at the beginning or middle of the sentence instead?

In Icelandic, adverbs of manner like snöktandi usually go at the very end of the clause they describe, just like in English. Moving it earlier (like Strákurinn fer snöktandi af sviðinu) is grammatically possible but places heavy emphasis on his sobbing rather than the action of leaving the stage.