Learn Dutch: Dan is de tegenstander aan de beurt. - Then it is the opponent's turn.

Because the sentence starts with the sequence word dan (then). In Dutch, the verb is almost always the second idea in a main clause (the "V2 rule"). When you put a word like dan at the beginning, the subject (de tegenstander) has to swap places with the verb (is) to keep the verb in the second spot.

How does aan de beurt translate to "turn"?

The phrase aan de beurt zijn is a fixed expression in Dutch meaning "to be one's turn". Literally, it translates to "to be at the turn". So instead of saying "it is his turn", Dutch speakers say "he is at the turn" (hij is aan de beurt).

Could I use toen instead of dan to mean "then" here?

Not in this sentence, because the verb is is in the present tense. Both dan and toen can mean "then", but toen is strictly used for a single specific moment or sequence in the past. Dan is used for the present, the future, or general repeating sequences (like "first my turn, then your turn").

What are the parts of the word tegenstander?

It is a compound word built from tegen (against) and stander (someone who stands). So a tegenstander is literally an "against-stander" — someone who stands against you in a game, debate, or conflict!