I'll give you a very basic explination sine I doubt you're intrested in all the fine details, also because its 1:30AM and I'm half asleep.
Deep down computers can only work with values of 0 and 1. One of these values is called a bit. Bits are grouped into collections of 8, these are called bytes. 1byte = 8bits.
Basicly systems handle the data they use in chunks. Large amount of data are passed around in little peices rather than in full, 32bit systems pass chunks of 32bit's (4 bytes) around so if you have 16byes of data to process the system will handle this in 4 seprate parts. (16 bytes = 128 bits of data, system cuts it up into 4 sets of 32)
Basicly with 64bit system data is handled in chunks of 64 rather than 32. A common myth is this doubles the speed because it handles twice as much in the same time but this is very much not the case, it simply speeds things up slightly becuase less time is taken to transefer/arrange the transfer of data. Compared to our previous case, the processor now hadles the 16 bytes of data in 2 parts rather than 4, this is more efficent and helps speed things up.
This is a very basic and probely not enterly accurate explination of it. In simple terms it makes the transfer of data more efficent within a system allowing for time to be spent on other things.