1. Did you like the restaurant (that) you ate at last night? Correct and natural. The word “that” is optional — I’d usually leave it out. 2. Did you like the restaurant where you ate last night? Correct, because the word "where" implies the preposition. Probably not as common as 1 in American English. 3. Did you like the restaurant at which you ate? Correct, but ridiculously formal sounding in today’s English. Anyone who claims version 3 is correct and version 1 incorrect is misinformed. Whatever you do, do not just say “the restaurant (that) you ate” — that would have an entirely different and nonsensical meaning.

Phil is absolutely right, The grammar behind it is that in adjective clauses we use "when" and "where" for time and place, but we can also use "preposition+which" instead of iether of them. If we want to use "that", the preposition needs to go to its original place. For example: January 12th is the day when I was born. January 12th is the day on which I was born. January 12th is the day (that) I was born on.