Learn Swedish: Mamman låter tonåringen sova sent. - The mother lets the teenager sleep late.

When you use låta to mean "let" or "allow" someone to do something, it works a lot like a modal verb. You follow it with the object (here, tonåringen) and then the bare infinitive of the action verb, which is sova without the infinitive marker att.

Doesn't låter usually mean "sounds"? How can I tell which meaning is used?

You are right that låta often means "to sound", as in det låter bra (that sounds good). However, when it is followed by an object and another verb in the infinitive (like tonåringen sova), it is acting as a causative verb meaning "to let" or "to allow".

Why is there a T at the end of sent?

Because it is being used as an adverb here to describe the verb sova. In Swedish, you typically create an adverb by taking the adjective sen (late) and adding a -t, which makes it look identical to its neuter form.

Why does the sentence use tonåringen (the teenager) instead of sin tonåring (her teenager)?

In Swedish, it is very common to use the definite form of a noun for family members or body parts when the relationship is completely obvious from the context. Saying tonåringen sounds more natural here than explicitly stating that it is her teenager.