The test book says only the first sentence is correct. I don't really understand why. Could anybody please help me with this?
This is not the theatre which he first played at.
This is not the theatre where he first played at.
In the absence of any other replies, I'll give you my opinion. I can't quote any rule, but I'll tell you how I understand it.
It's not correct to use the preposition "at" with "where", because they both indicate place, and the preposition is redundant.
I can remember teachers and parents correcting us as youngsters when we asked our friends, "Where are you at?"
"Where are you?" is all you need to ask. Likewise, "... where he first played." is complete without the "at".
The reason for the "at" in the first sentence is that we can't play a theatre as we play a piano, but we can play in or at one.
This, notwithstanding the fact people will say things like, "He played The Albert Hall." In this case, the missing "at" is said to be understood. So, one could also say, "This is not the theatre which he first played." This is less formal usage and could be classed an error by some.
This kind of question is very popular in my country.
We have an explanation as below:
"This is not the theatre which he first played at." can be divided into two sentences; "This is not the theatre." and "He first played at the theater." You combine these two into one by using "which" instead of "the theatre." Therefore you need "at."
"This is not the theatre where he first played." can also be divided into two; "This is not the theatre," and "He first played there (at the theatre)." You combine them by using "where" instead of there (at the theatre).