The phrase "wear ___ out" has two meanings. One meaning is to use something until it's worn out:
I love this jacket. I'll keep wearing it until I wear it out.
But the meaning of "wear ___ out" that's used in the example above means to wear something to go out somewhere. In the situation at the top, wearing the costume out means going to a party or parade.
Notice that for the first meaning of "wear ___ out", you can also say "wear out ___":
It takes a long time to wear out a nice pair of shoes like those.
But you can't say "wear out ___" when you mean "wear something to go out".