From Topsy and Tim, Series 1, episode 2.
-Right, that's me, I'm done. See you tomorrow.
-Bye, thanks, Derek.
What exactly did "that's me" mean in this context?
e. it means essentially the same as "I'm done". "that's me" may derive from a shortening of "that's me finished" or similar.
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In this case "that's me" is a casual conversational way of saying "I've finished (doing something)", i.e. it means essentially the same as "I'm done".
"that's me" may derive from a shortening of "that's me finished" or similar.