What does request_mem_region() actually do and when it is needed?

Using request_mem_region() and ioremap() in device drivers is now deprecated. You should use the below "managed" functions instead, which simplify driver coding and error handling:

devm_ioremap()
devm_iounmap()
devm_ioremap_resource(), Takes care of both the request and remapping operations

Look at slide 289 of bootlin company training course.

request_mem_region tells the kernel that your driver is going to use this range of I/O addresses, which will prevent other drivers to make any overlapping call to the same region through request_mem_region. This mechanism does not do any kind of mapping, it's a pure reservation mechanism, which relies on the fact that all kernel device drivers must be nice, and they must call request_mem_region, check the return value, and behave properly in case of error.

So it is completely logical that your code works without request_mem_region, it's just that it doesn't comply with the kernel coding rules.

However, your code doesn't comply with the kernel coding style. And additionnally, there is an existing infrastructure to handle GPIOs, named gpiolib, which you should use instead of manually remapping your GPIO bank registers. Which platform are you working on ?