Learn German: Warum muss dieser Angeber ständig angeben? - Why does this show-off constantly have to show off?

Yes! Angeber (show-off) is the noun form of the verb angeben (to brag or to show off). German frequently creates nouns for people who do a specific action by taking the verb stem (angeb-) and adding -er, just like in English.

Why is angeben at the very end of the sentence?

This is due to the modal verb muss (must or has to). Whenever you use a modal verb to express ability, obligation, or necessity, it pushes the main action verb to the very end of the clause in its full infinitive form.

I thought angeben was a separable verb. Why isn't the an- separated here?

It is indeed a separable verb! However, because the modal verb muss forces it to the end of the sentence in its infinitive form, the prefix stays attached. It would only separate if it were the main conjugated verb, for example: Er gibt ständig an (He shows off constantly).

Could I use immer instead of ständig?

Yes, immer (always) would work grammatically, but ständig (constantly) makes the sentence more expressive. It emphasizes the persistent, continuous, and often annoying nature of the bragging.