What does the phrase rack your brains mean?

"Rack your brains" means to think very hard about something, often in an attempt to solve a problem or remember something. It implies a concentrated effort to come up with an answer or solution.

Here's an example:

"I've been racking my brains all day trying to remember where I put my keys."

The phrase is a metaphor, drawing on the image of "racking" (stretching, straining) a physical object, like a rack. In this context, the "rack" refers to the mind, suggesting that you are pushing your mental capacity to its limit.