lui or le?

The verb écouter takes a direct object, hence it is J'aime l'écouter.

Whether a verb takes a direct or indirect object (some take both, some change meaning with one or the other) needs to be learned together with the verb. There's no hard and fast rule. For example télephoner takes an indirect object while appeler uses a direct object:

J'ai téléphoné à Marie. -- I phoned Marie (Marie is the indirect object).
J'ai appelé Marie. -- I called Marie (Marie is the direct object).