Let's put it this way:
32-bit means 4 groups of 8 bits. So an address space of 0 to 4,294,967,295 is possible (from a 32-bit integer that is all zeroes to a 32-bit integer that is all ones).
64 bit means 16 groups of 8 bits. This is MUCH larger, an address space of 0 to 9,223,372,036,854,775,807! (That's 9 quintillion, 223 quadrillon, 372 trillion, 36 billion, 854 million, 775 thousand, 807 bytes!)
While this number is massive and we may not achieve it in our lifetimes for a variety of reasons, there's a lot more space to work with, so a lot more stuff can be represented in memory.
While 64-bit computing is the new standard, 32-bit computing won't go anywhere anytime soon, not unless we develop 128-bit computing...