Does the cubic capacity of a dirt bikes engine have to do with the size of the bike?

The size (height and weight) has nothing to do with the engine size. "cc" is shorthand for cubic centimeter. Displacement measures how many cubic centimeters of the air-fuel mixture moves through one rotation of an engine cycle (each piston moving from top to bottom). For example, if one rotation of the pistons moves 125 cc of the fuel-air mixture through the system, then you have a 125 cc engine.