Learn Dutch: We zien de muzikant op het festival optreden. - We see the musician perform at the festival.

Because zien (to see) is a verb of perception. Just like in English (We see the musician perform), Dutch verbs of perception like zien, horen, and voelen are followed by a bare infinitive. This means you use the whole verb without adding te.

Why is optreden at the very end of the sentence instead of right after de muzikant?

In Dutch, when you have a conjugated verb like zien paired with an infinitive like optreden, the infinitive is pushed to the very end of the sentence. The extra details, like the location (op het festival), are sandwiched in between.

Optreden is a separable verb. Why isn't it split up here?

Separable verbs only split when they are the main, conjugated verb of the sentence (for example: Hij treedt op). Here, the conjugated verb is zien. Because optreden is just an infinitive sitting at the end of the sentence, it stays whole.

Why do we use op to mean 'at' the festival?

In Dutch, public events like a festival, concert, or party usually take the preposition op. You say you are op een feest (at a party) or op het festival (at the festival).