To "get to do something" means to be allowed the privilege of doing something. It has to be a privilege. You don't "get to do" something that you don't want to do (unless you are saying it sarcastically, as J.K. pointed out). For example, a mother might say to her child "you have eaten all of your spinach, so now you get to eat your dessert".

In this context, "(they) get to" means "(they) are allowed to." The speaker (I'm guessing it must have been Barney) is being sarcastic. He doesn't think wearing a wig is a privilege. He thinks it's silly.

Get to means you are allowed to do that thing. Other examples: I get to wear jeans or casual clothes on Fridays at work because my work does casual Fridays. I get to have the day off from work on Sundays. I get to driving a car when I turn 16. I teach English classes. Let me know if you are interested in booking a class.