Use this expression when you want to complain about something. "The thing about ___ is...." introduces an idea that you want to complain about:
The thing about the current political administration is that they're all talk and no action.
The one thing about Lacey that annoys me to no end is that she's never had any hands-on experience, yet she seems to think that she knows everything about it.
You can also use "The thing about ___ is ___" in other situations where you're not complaining. You can also use it to introduce a positive or interesting point about something. For example:
The thing about my Uncle Frank is that, once you get to know him, he's really nice and very loyal.
The thing about college is, you don't really learn very much that you couldn't learn on your own by reading a few books.