grammar - Why is this sentence grammatically correct?

The Why part is easy. It's grammatically correct because it follows grammatical rules correctly.

But there are a lot of English grammatical rules, and this sentence

• For music isn’t just something nice to listen to.

makes use of quite a few. This sentence has three clauses, two of them reduced clauses.

Plus, it starts with the coordinating conjunction for,
which indicates that it's conjoined to some unknown previous sentence.
Let's ignore the for part here, because we don't know what the previous sentence was.
(Though @BillJ's guess is as good as any)

• Music isn’t just something nice to listen to.

Just means only (in its sense 'of small importance'), so it's predictable.
And it's directly negated by the negative isn't, so the sense is reversed: 'of great importance'.
Again, predictably. And with the same essential structure as

• Music is something nice to listen to.

Let's start by putting back in the optional markers that have been optionally deleted

• Music is something that/which it is nice for Indef to listen to.

In this case, the chunks that have been deleted include

1. the relative pronoun that or which (inserted by the rule of Relative-Formation)


2. the dummy subject it (inserted by the rule of Extraposition)


3. the auxiliary verb is (required for all predicate adjectives like nice)


4. the infinitive subject marker for (part of the infinitive for..to complementizer)


5. the indefinite infinitive subject (the person(s) doing the listening: Indef).

Note that the sentences with and without the deleted chunks mean the same thing.

It's also possible to unwind other rules here. The dummy it, for instance, comes from Extraposing

• For Indef to listen to music is nice for Indef
  ==> Extraposition ==>
  It is nice for Indef to listen to music
  which gets rid of the double phrase for Indef; it's positioned to modify either way.

There's more to say; this sentence has been done a lot of things to.
But I'll stop here.