Learn Spanish (Latin America): Me enoja que el niño no escuche. - It angers me that the boy doesn't listen.

We use the subjunctive because the first part of the sentence expresses an emotion (me enoja). In Spanish, when you express how you feel about an action done by a different subject, the second verb (after que) must be in the subjunctive mood.

Does me enoja work like the verb gustar?

Exactly! Just like me gusta means 'it pleases me,' me enoja means 'it angers me.' The subject of the first part of the sentence is actually the whole idea of the boy not listening (que el niño no escuche). That entire idea is what does the angering to you (me).

In English we might say 'It angers me for the boy not to listen.' Can we drop the que and use the infinitive escuchar in Spanish?

No, you must use que and the conjugated subjunctive verb (escuche) here. You only use the infinitive (escuchar) if there is no change in subject. For example, me enoja no escuchar means 'It angers me that I do not listen.' Since the anger is felt by you but the listening is done by the boy, you have to use que plus the subjunctive.

What is the difference between me enoja and estoy enojado?