Learn Hardcore Japanese: zyugyouryou no siharai no tame ni, ginkou ni ikimasu.

Each の links two nouns into a larger noun phrase. The first の connects 授業料 (tuition) and 支払い (payment) to form “payment of tuition.” The second の links that entire noun phrase (授業料の支払い) with ため (sake, purpose), yielding “for the purpose of paying tuition.” You need both to express those two relationships.

What does ために mean here, and why is it used instead of just ため?

ため by itself is a noun meaning “sake” or “benefit,” but when you want to express purpose (“in order to…”) you add に to make it an adverbial phrase. So ～のために means “for the sake of…,” “in order to…,” and it indicates the goal or purpose of the action that follows.

Could I rephrase this sentence using the verb 支払う instead of the noun 支払い?

Yes. You can say:
授業料を支払うために、銀行に行きます。
Here 支払う (to pay) is in its plain dictionary form plus ために, directly expressing the purpose. It’s more common in speech, whereas ～の支払いのために is a bit more formal or written style.

What’s the difference between 授業料を支払うために and 授業料の支払いのために?