Which one is correct?
- I don't need to know where you work at.
- I don't need to know where you work.
Could you also please tell me about this rule is called in grammar so I can learn more about it?
The copyeditor in me observes that your at there adds nothing to the sentence: leave it out.
It’s also something of a shibboleth whose use will mark you as “poorly educated”, because these things are expected to have been corrected in the course of proper schooling. The same is true with questions like “Where are you at?” Here again that at at the end adds nothing — or at least, nothing good — and should be omitted.
Once upon a time, surely aeons ago now, there was something of a hipster phrase about “where it’s at”, referring to some location or event or even style that was supposed to be “really happening”. Unless you’re trying to faithfully recreate the special slang from that period in history, I would avoid it.
When we use the Interrogative Pronoun "Where," we normally don't include the Prepositions, whether it's in the form of Questions, Noun Clauses or Noun Phrases.
Where are you going? (no "to")
Where do you live? (no "in")
Compare these wit:
WHICH restaurant are you going TO?
WHAT city do you live IN?
Except when we ask about origins:
Where are you FROM?