Being able to "carry a tune" means that you are able to sing pretty well. Here's how to use "carry a tune":
A: Are you a good singer?
B: Yeah, I can carry a tune.
In this example, the second person is being a little modest. That's because "I can carry a tune" doesn't mean that you're a good singer. It just means that you can sing decently well.
If someone is a really bad singer, say that they "can't carry a tune":
Allen, a good singer? No, he couldn't carry a tune to save his life?