The main reason of using 64-bit word size over 32-bit word size is speed, on CPUs that manipulate words sequentially. Doubling the word size can hope to about double the number of elementary bitwise operations performed in a given amount of time, on a 64-bit CPU like one using the AMD64 instruction set.
As an aside, for constant number of words in the internal state of a an algorithm (like a hash), the number of bits in the state is twice as large, which can improve the cryptographic resistance (if other conditions apply; in particular, at least twice as many bit operations); and allows a larger result.
Accordingly, for large input, SHA-512 is typically faster than SHA-256 on the same 64-bit CPU, even though SHA-512 uses 25% more rounds, and has a state with twice more bits, making it arguably safer.
Update per comment: word size has no direct impact on security. As an illustration, it is possible to reformulate SHA-512 as using 32-bit words, with identical result thus security; compilers for 32-bit CPUs do this automatically.