All calculations take place in the registers. When you're adding (or subtracting, or whatever) variables together in your code, they get loaded from memory into the registers (if they're not already there, but while you can declare an infinite number of variables, the number of registers is limited). So, having larger registers allows you to perform "larger" calculations in the same time. Not that this size-difference matters so much in practice when it comes to regular programs (since at least I rarely manipulate values larger than 2^32), but that is how it works.
Also, certain registers are used as pointers into your memory space and hence limits the maximum amount of memory that you can reference. A 32-bit processor can only reference 2^32 bytes (which is about 4 GB of data). A 64-bit processor can manage a whole lot more obviously.
There are other consequences as well, but these are the two that comes to mind.
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