Basically
Here is the best layman explanation I can muster.
Information moves in 8 wires, also known as 8-bit data. 8bits is a byte of a single pice of data.
A 32 -bit CPU can take in 4 pieces of data at once, 4 pieces * 8 bits = 32-bit.
A 64-bit CPu can take in 8 pieces of data at once, so to really sum it up the difference is the amount of data that the processor can receive in a single clock beat.
A 32-bit 2.4Ghz CPU can read 76,800,000,000 bits or 9,600,000,000 bytes (characters) in a second.
A 64-bit 2.4Ghz CPU can read 153,600,000,000 bits or 19,200,000,000 bytes(characters) in a second.
To sum it up, the 64-bit CPUS, even with the same Ghz clock rating can process twice the information that a 32-bit CPU can.