I saw a hilarious TV clip about why this is wrong.

Problem:
The number of the verb does not match the number of the subject.

Explanation:
The pronoun “none” means not one.

The pronoun “one” is singular and takes the third-person, singular form of “work”, which is “works”.

Seeing this, we get “Not one of them works.”

Converting “Not one” back to “None” gives us the solution.

I believe that the problematic sentence is an example of speakers and writers being distracted by the nearness of the verb to the pronoun “them”, which is plural.

Solution:
“None of them works.”