Windows XP 64-bit and above all can run PC games galore easily. With enough installed memory, 64-bit versions of Windows will load system components into higher extended memory and away from the lower 2GB where legacy games will run. MIcrosoft calls this Windows on Windows and it has been done when 16-bit moved to 32-bit earlier.
Our first 64-bit CPU was the AMD Athlon64 which provided the goal of more memory capacity with full backwards compatibility for 32-bit binary code. AMD supports mixed 32-bit and 64-bit code. Intel in a cross license deal uses the same architecture. Now both AMD and Intel both have excellent support for 64-bit systems.
By freeing up lower memory, the game has much more memory to work with. For example, the game Left 4 Dead 2 etc. uses main memory easily and can even allocate 1GB of video card memory. Other games experience similar performance gains.
As the amount of memory rises from 4GB upwards, 64-bit Windows becomes much more efficient. Remember that video card VRAM is memory mapped into the address space so the expanded address space makes it easier to cope with video cards with more than 512MB VRAM.
The 64-bit video driver can map the game calls to extended memory easily. This makes the game run better as there is much room for memory generally.
The vast majority of 32-bit games work fine with 64-bit Windows. There are a few that do not work, mostly due to DRM scheme used rather than the game itself.