What exactly does 30 inch clearance mean? We are getting ready to remodel our kitchen and the existing circuit breaker box is in the way of where we want the cabinets to go. We would like to hide it in one of the new cabinets with a cutout in the back and not put anything in the cabinet. Is this allowed?

What exactly does 30 inch clearance mean? We are getting ready to remodel our kitchen and the existing circuit breaker box is in the way of where we want the cabinets to go. We would like to hide it in one of the new cabinets with a cutout in the back and not put anything in the cabinet. Is this allowed?
ABSOLUTELY NOT.
A panel in a kitchen cabinet is one of the worst violations most of us have seen. And many of us have (if you can believe it) seen it more than once.

The clearances are 30" wide and 36" deep (in front of the panel), and 6-1/2' high.
The panel does not need to be centered in the 30" wide space.

Strongly recommend you follow Stan's recommendations...
They are correctly stated right out the Electrical Code book.
Maybe relocation of the panel would be more acceptable?