Both of them are correct.

Va-t-elle demander les explications au professeur ?

Oui, elle va demander les explications au professeur.

Demander quelque chose (something) à quelqu’un (someone).

Oui, elle va les demander au professeur.

The pronoun les replaces here les explications.

Oui, elle va les lui demander.

Here, in addition, the pronoun lui replaces au (=à+le) professeur.
Note that les precedes lui.

As a remark I find that repeating the whole sentence in the answer is very formal/scholar. In real life, whenever the answer is not a plain and simple “oui”, there’s a big chance that “Oui, elle va les lui demander” will be used, possibly with even less pronouns/complements as these are implied by the context. So “Oui, elle va demander” could be used. But the original question being an exercise, this short answer wouldn’t fit the requirements. (Thanks @Laurent S.)