I would like to propose an answer to this question, since 15 year ago I was asking it to myself and was thinking that orbiolds are useless. I read your question (maybe wrongly) as a question in mathematical psychology (or just in psychology). At the present moment I use orbifolds very often. And there was a turning point for me, when I understood one particular statement about orbifolds that made geometric sense to me.

Statement. An orbifold of negative curvature is a global quotient.

For some reason, once I got this statement, I decided that orbifolds are cool. But I know that this is very personal, so I don't expect that this particular statement will necessarily be interesting for you.

Also, there is a very nice example of an orbifold mentioned Thurston's notes on 3-manifolds - the Barber shop.