Hola Jenk,
This "me" here in this sentence is what is called "dativo ético". We sometimes use indirect object pronouns (me, te, le, nos, os, les) to indicate who the action of the sentence is indirectly affecting somehow. "Me" here is saying who is being affected. So the comedian in the video is trying to make a joke about the state of the stage where she is, as it if was messy (como una leonera) and it somehow that affects her.
Have a look at other sentences using the pronouns in the same way:
My little boy doesn't eat anything
The "me" here again is making a point about how this not eating is affecting the mum (the speaker). You could perfectly say the same sentence without "me", but with it, it will add that nuance to the meaning.
What are you saying? your kitten died?
Again, here, "te" reflects who is affected by what happened.
I hope this helps.
Interesting stuff!
Un saludo
Inma