Learn Hardcore Russian: Я слышу музыку вечером.

Because it’s the direct object of the verb слышу (“I hear”), so it must be in the accusative case. The noun музыка is feminine; in the accusative singular, -а changes to -у: музыка → музыку. Note: with many inanimate masculine nouns the accusative equals the nominative (e.g., слышу звук).

What’s the difference between слышать and слушать?

слышать = to hear (perceive sound, not necessarily on purpose): Я слышу музыку от соседей. “I hear music from the neighbors.”
слушать = to listen (actively, intentionally): Я слушаю музыку вечером. “I listen to music in the evening.” If you mean you intentionally put music on, use слушать, not слышать.

Does вечером mean “this evening,” “in the evening,” or “in the evenings”?

вечером most often means “in the evening” of a day under discussion; in today’s context it reads as “this evening/tonight.”
For habitual/repeated action, Russian usually prefers по вечерам or вечерами (“in the evenings”). Examples:
Я слышу музыку по вечерам / вечерами. “I hear music in the evenings (habitually).”
Сегодня вечером highlights “this evening (today).”

Why is there no preposition before ? Why not ?