The thief is Pierrot.

The people around are telling the storyteller that "he" (Pierrot) has stolen the purse, but the storyteller misinterprets this as pointing to some arbitrary agent.

Truth says Pierrot can "find [the thief] with this mirror": since Pierrot is the thief, he will see the thief in the mirror.

Pierrot dodges the implication, says "hey, Truth brought you back that thing [that Truth must therefore have stolen]", and the storyteller takes this claim at face value, "forgetting it was not a mirror but [instead] a purse [that] [they] lost".

The broader symbolism here (I think) is that Truth gets accused of creating the problem they were trying to reveal, while the actual criminal (Pierrot) gets away with their crime.