Learn German: Ziehen Sie Ihre Schürze an, während Sie das Frühstück machen! - Put on your apron while you make the breakfast!

Because während (while) is a subordinating conjunction. It introduces a dependent clause, which forces the conjugated verb (machen) to the very end of that clause.

Does während always mean 'while' as in two things happening at the same time?

In this sentence, it shows simultaneous actions: wearing the apron and making breakfast. However, während can also show a contrast, similar to 'whereas' in English (for example: I like tea, whereas you like coffee). Here, the context clearly points to time.

I thought anziehen was a separable verb. Why are ziehen and an split up here?

That is exactly how separable verbs work in the imperative (command) form. In a formal command, the main part of the verb comes first (Ziehen), followed by the pronoun (Sie), and the separable prefix (an) gets pushed to the very end of the main clause, right before the comma.

Why is it Ihre Schürze and not just Ihr Schürze?

Schürze (apron) is a feminine noun (die Schürze). Because it is the direct object being put on, it is in the accusative case. For feminine accusative nouns, the formal possessive pronoun Ihr (your) takes the -e ending, becoming Ihre.