Learn Hardcore Polish: Biblioteka naliczy opłatę.

Because naliczy is from the perfective verb naliczyć. In Polish, perfective verbs do not normally have a true present-tense meaning. Their present-looking forms usually refer to the future.

So:

naliczać = imperfective, to be charging / to charge repeatedly
naliczyć = perfective, to charge / assess / add up as a completed action

That means:

Biblioteka nalicza opłatę. = The library is charging / charges a fee.
Biblioteka naliczy opłatę. = The library will charge a fee.

So even though naliczy looks like a present form, it means will charge here.

What exactly does naliczyć mean in this sentence?

Here naliczyć means something like:

to charge
to assess
to apply
to add on

In this context, it usually means that the library will officially add a fee to your account, for example for:

a late return
damage
some service charge

So Biblioteka naliczy opłatę is a natural way to say that the library will impose or add a fee.

Why is it opłatę and not opłata?