It's from the second episode of the tenth season of Friends. Here is the scene. It's at 1 minute and 49 seconds.
Chandler: I'd love to, but I gotta get back to talking to your parents. They're telling us all about how they adopted you.
Owen: What?!?
Chandler: What?
Owen: I'm adopted?
(Chandler tries to come up with a good answer.)
Chandler: I got nothing.
In this context, "I got nothing" means "I don't have anything to say". It could mean that you choose not to add anything to a discussion, but often in this kind of context, it means that you have tried to think of something to add but were unable to.
As you point out, this is his response after he tries to come up with a good answer.
There is also an essentially identical remark in some gambling card games, such as various forms of poker where players' cards are not all visible: namely, while there can be bluffing and non-disclosure and so on through the play of a hand, at some point there will be "a reckoning", and people with poor hands will often not even bother to show how poor their hand was (in part to avoid divulging information about what they'd consider a partly viable or bluffable hand...)
