Paging is used in the Virtual Memory (VM) system. VM is an address space the OS provides for processes that sort of acts like RAM but is stored on disc (HDD/SDD) rather than in memory. This is for the exact reasons mentioned by gamerk316. I believe, although don't quote me on this, the maximum size of the virtual memory is around 3 times the size the system RAM.
The virtual memory is partitioned up into equal sized sections, these are the pages. Usually around 4k-8k in size. If the pages are too large then the swapping process takes too long, if the pages are too small then you will have to make swaps too often.
Some of the virtual memory pages will be stored in physical memory (page frames), these are the ones in use by a process at that moment in time, or the ones that have been used recently. All the rest of the pages are on disc. If a running process tries to reference a memory address that is not stored in physical memory, i.e its trying to access one of the pages that has been swapped out to disc, you get a page fault. So one of the pages in memory gets swapped out for the page you need that is on disc.
On to the swapping out process. Your OS wants to try and swap out a page that is not going to be needed again for the longest period of time. Pages have a few attributes associated with them that can help this process, although accuracy is difficult to achieve as it is near impossible for the OS to predict the execution of a program. You have the following attributes:
Present/absent: true if this page is currently stored in physical memory (i.e can this page be swapped out?)
Page frame number: The physical page frame this page is stored in (if in physical memory where is it stored?)
Modiﬁed: Has the data been modiﬁed since it was read from disk? (AKA a dirty bit, these changes have to be written to disc so they take slightly longer to swap out)
Referenced: Has the data been referenced (recently)
Protection: Is the data read/write/executable?
There are a bunch of page replacement algorithms but essentially they look for the following things when deciphering a page to swap out:
The page is in memory (present)
If there are several pages that can be swapped out, look for clean (un-modiﬁed) pages that will not need to be written to disk
(Poor page replacement can result in thrashing — processes that rapidly and repeatedly have page faults)
In the context of this question, the reason some people suggest disabling page file checks is that Windows can be inefficient when using the pagefile. So if you have a high end pc, with lots of RAM you can just disable it since RAM is a lot faster than your hard drive. By disabling it, you are forcing Windows to keep everything in much faster RAM all the time.
The downside to turning of the use of pagefile is that once your system has used all of the available RAM, your programs are going to start crashing. This is because there is no additional virtual memory for Windows to allocate. Is some cases your entire computer will crash or become very unstable.
I can't imagine disabling page file checks will improve game performance all that much as swapping is handled by the Memory Management Unit that is built in to the hardware. Considering the price of this tweak is a potentially unstable system that could be prone to crashing I think you'd be better off tweaking the graphics settings as opposed to doing something like this. Unless you have 16GB of RAM or something.
EDIT: I hadn't realised how long i'd babbled on about this until I posted my answer... TL;DR check last 3 paragraphs