Meaning/Usage: Have knowledge of every detail of something.
Explanation: There are two sides of a flat object like a coin. There is also an inside or outside to many objects. This phrase is saying you know both sides, which indicates knowing everything about it.
Ask Mary to help you with your homework. She knows calculus inside out.
If you got the answer from Matt, I would trust it. He knows that stuff inside out.
A:
I'm having trouble with my car. I can't seem to figure out what's wrong with it.
B:
You should ask James about it.
A:
Does he know a lot about cars?
B:
He's a mechanic. He knows the car engine inside out.