The "bitness" of a processor is defined by the width of the general purpose registers. The registers are bits of memory on the processor that store data that is being used. Each general purpose register stores 1 data type (no searching is required, there is no latency). They are just what they sound like, general purpose. They are capable of storing any of the 3 (on current x86 MPU's) data types the MPU handles. Integer, floating point, or memory address.
Basically, expanding the general purpose registers means you can handle bigger numbers. In 32-bit MPU's, you can store (and thereby handle) 32-bit data types. Such as a 32-bit integer or a 32-bit memory address. I mentioned only integer and memory addresses because floating point data have their own dedicated register stack (which are 80-bit wide in modern x86 MPU's). The general purpose registers (8 on x86 MPU's and 16 64-bit ones on Hammer) are almost never used (why am I saying almost, never) used to hold floating point information.
So basically, having a "64-bit" MPU would mean that you can handle 64-bit data types. Such as a 64-bit integer or a 64-bit memory address. So if an integer you want to do work on is ever bigger than 32-bit (which is the range of 2.7 million or negative 2.7 million), you won't have to split the number up into smaller parts, work on them separately, and then put them together (very slow). There is also the fact that if you need a memory address bigger than 32-bit, you can do it. The amount of memory the MPU can address is the reason for the 4GB limit (in terms of memory) you can put in a computer based on a 32-bit MPU.
So basically, the move to 64-bit by itself would provide 2 advantages:
1. The ability to handle bigger integers without significant loss to performance.
2. The ability to address more than 4 GB of memory.
Now, there are a few things I should point out. First of all, Intel's MPU's can use 36-bit virtual addressing. This isn't as convenient as flat addressing (as the OS has to be aware of it and switch modes) but it does allow more memory to be addressed.
Secondly, no consumer-level application ever uses that big of an integer. Most applications nowadays are FP intensive anyway (especially games). However, in workstation/server level applications like database software, large numbers can help significantly.
"We are Microsoft, resistance is futile." - Bill Gates, 2015.