I came across this line; "I'm happy that you're happy" in movie.
What does 'that' mean here? Does 'that' in here have the same meaning as 'when'?
I in no way thought that 'that' would address this kind of meaning.
"That" is a conjuction. It links two clauses, and it introduces a subordinate clause. The subordinate clause gives the reason or cause of the main clause. So "that" is close in meaning to "because".
I'm happy that she's happy. (the subordinate clause "she's happy" gives the reason why I'm happy.)
Unlike "when" there is no conditional sense. If "I'm happy when she's happy" then "Am I happy now" can't be answered, there isn't enough information. On the other hand "I'm happy that she's happy" implies that "I am happy now" and "She is happy now"