This phrase means "don't worry about it". That meaning comes from the fact that, when people worry, they sweat.
You can use "Don't sweat it" to calm down someone who's worried:
A: I'm so nervous.
B: Ah, don't sweat it. You'll do fine.
You can also use it as a reply when someone thanks you:
A: Thank you so much for your help.
B: Don't sweat it.
Be a little careful using "don't sweat it" in this way, though. It sounds like something that a person says to a close friend or someone who's lower in status, like an employee.