Learn Hardcore Norwegian: Han vil snakke flytende.

Because vil is the finite verb in the sentence. After a modal verb like vil, the next verb stays in the infinitive form, so you use snakke, not snakker.

Compare:

Han snakker. = He speaks.
Han vil snakke. = He wants to speak / He will speak.

In Norwegian, you normally have one finite verb per clause, and here that verb is vil.

Why is there no å before snakke?

After modal verbs, Norwegian usually uses the bare infinitive, without å.

Common modal verbs include:

kan = can
skal = shall / will / is going to
vil = want to / will
må = must
bør = should

So:

Han vil snakke is correct
Han vil å snakke is not correct

What exactly does vil mean here?

Vil can often mean either wants to or will, depending on context.

So Han vil snakke flytende could be understood as:

He wants to speak fluently
He will speak fluently

In real life, context usually makes the meaning clear. Since the learner has already been shown the intended meaning, that is the one to follow here. But it is useful to know that vil can be ambiguous by itself.