Learn Italian: Lui se ne vanta sempre, ma non se ne occupa mai. - He always brags about it, but he never deals with it.

When the reflexive pronoun si combines with the particle ne, the i in si changes to an e, making it se ne. This happens because both vantarsi and occuparsi are reflexive verbs, and this standard vowel change makes the double pronoun cluster easier to pronounce.

What exactly is ne replacing in both halves of this sentence?

The particle ne often replaces phrases starting with the preposition di. Both verbs used here require it: vantarsi di (to brag about) and occuparsi di (to deal with or take care of). Therefore, ne stands in for about it in the first half and with it in the second.

Why did they include the pronoun Lui if Italian usually drops subject pronouns?

Including Lui here adds emphasis. While grammatically optional, stating it explicitly points a finger directly at him, highlighting the frustrating contrast between his constant boasting and his total lack of action.

Why does the second half of the sentence use both non and mai?