Learn German: Du musst mir bitte zuhören. - You must listen to me, please.

The verb zuhören (to listen to) always requires the dative case for the person you are listening to. Because of this, we must use mir, which is the dative form of ich (I), rather than the accusative form mich. A helpful trick: the prefix zu- comes from the preposition zu, which always takes the dative case!

Why is zuhören at the very end of the sentence?

This happens because of the modal verb musst (must). When you use a modal verb in German, it takes the standard second position in the sentence, and it pushes the main action verb (here, zuhören) to the very end in its dictionary form.

Isn't zuhören a separable verb? Why isn't it split up into zu and hören here?

You are right that zuhören is a separable verb! In a normal present tense sentence, it splits up, like in Ich höre dir zu (I listen to you). However, because the modal verb musst forces the main verb to the end of the sentence in its full, unchanged form, the two parts of zuhören stay glued together.

Does bitte always go right before the verb at the end?

Not necessarily! The word bitte (please) is quite flexible in German. It usually sounds best after the main pronoun objects, which is why it comes right after mir here. You will often see it placed in the middle of a sentence rather than at the very end like we do in English.