BARs are part of the PCI spec, they're the way for firmware and OS to determine the size of the memory or i/o window the card controls as well as the way for the system firmware (usually) to pick what address range the card should use.

In the old days, they were a fixed size. If your GPU has 16 GB of ram and you want to access all of it via memory addresses, you'd need a 16 GB BAR ... but lots of (older) systems wouldn't be able to map in a large BAR because of a lack of address lines (or wahtever), so GPUs had stayed with 256MB for VRAM access because it was compatible. With a smaller BAR than the VRAM, you have to use some sort of windowing / paging setup. Resizable BAR lets the BAR start small so older systems will work, but grow larger with capable systems.

Much better than having a jumper to set the BAR to big or small, and you can skip VRAM window management.