Learn Icelandic: Maðurinn lætur nuddarann nudda bakið. - The man lets the masseur massage the back.

In the causative construction, the verb láta takes an accusative object for the person being told or allowed to do the action. The masculine noun nuddari (masseur) becomes nuddara in the accusative, and adding the definite article gives us nuddarann.

Does lætur mean "lets" or "makes" in this context?

The verb láta can mean both "to let" and "to make" (or "to have someone do something"). While the literal translation is "lets", in this context it most likely means he "has" the masseur massage his back. You rely on context to tell whether it is permission or a request/command.

Why doesn't the sentence say bakið sitt (his back)?

In Icelandic, it is very common to just use the definite article for body parts when it is obvious whose body it is. So bakið (the back) naturally means "his back" here. You do not need to add a possessive word unless you want to specifically emphasize whose back it is.

Why is bakið also in the accusative case?

Because it is the direct object of the infinitive verb nudda (to massage). Even though the sentence is governed by láta, the action of massaging is what directly affects the back, and the verb nudda requires an accusative object.