Good first question. :)
1) de separates le plus and musées; musées is not really the noun introduced by le, even if it looks like it. As a result, le is probably best described as not agreeing with any other word.
2) As the other answerers wrote, le plus de is always correct. However, as Lambie writes below, it may be informal compared to fuller formulations such as le plus grand nombre de.
Difficulties with agreement are always going to involve the question: "Which elements are agreeing?" This question comes up whether you're talking about conjugating verbs, modifying adjectives and participles, choosing determiners...
Here, you would begin by asking about le. Which word is it introducing?
The English translation you gave suggests that museums would be the nearest noun:
...the most museums.
But while this is a good translation, it's not quite the grammar of the French:
...le plus de musées.
This structure, given away by the presence of de, is a common one that usually links two nouns:
Une boîte de farine
Une tasse de thé
Or a quantity and a noun:
Beaucoup de farine
Plus de musées
And in both cases note that the noun itself loses its own determiner:
De la farine
but Une boîte de farine
but Beaucoup de farine
This suggests that in your textbook's sentence, the determiner for musées would have looked something like this:
Des (de + les) musées
but Plus de musées
All this is to show that the le is not for musées. Its les disappears in this structure.
So what is the le for in this phrase? I guess there are two ways the analysis could go.
Maybe you could say that plus is short for plus grand nombre, for example, in which case le agrees with nombre (and we would have a noun + noun nombre de musées instead of quantity + noun plus de musées).
But that explanation is a bit too ad hoc for my taste. We could also do it grammatically and start with the rule that the comparative is plus, and the superlative is made by adding the definite article:
Elle est intéressante
Elle est plus intéressante
Elle est la plus intéressante
When we're comparing an amount of something, we still use plus:
Paris a plus de musées
And we still make it superlative with the definite article:
Paris a le plus de musées
But we notice two interesting things...
First, that neither with comparing adjectives nor with comparing quantities is this le really a normal determiner. After all, even in "la plus intéressante" or any other comparative phrase with an adjective, the determiner is not followed by a noun as you would expect it to be. Probably the determiner just picks up the gender and number features of the next item in the same constituent or "block" of the sentence.
The second (key!) thing we notice is that these are not really the same plus. One is followed by an adjective, and the other by de. Even without further analysis of what they are, that tells us they behave differently.
And if we skip ahead a bit in linguistics we find that de begins a new constituent, the prepositional phrase. That's enough to separate le from the noun, meaning le won't have an immediate neighbour from which it would borrow gender and number features. So it retains the unmarked form, le.