Conventional video cards are limited to a large degree by memory bandwidth. They usually solve this by adding more, faster memory. PowerVR, with TBR, has tried to solve it a little more directly. Usually the video card will load any texture on screen, even hidden ones. Textures are what the vast majority of memory bandwidth gets used for, so if you can cut down on the number of loaded textures, you cut down bandwidth usage. Tile based rendering involves figuring out which textures are hidden by other objects, and only loading visible ones. It also tries to avoid using external memory by breaking the screen into small tiles so calculations can be done on-chip.
PowerVR has implemented tile based rendering on its kyro I and kyro II chipsets, both of which can reach speeds similar to non-TBR chipsets with less processing power, making these cards significantly cheaper. Neither, unfortunately, has caught on against fierce competition from ATi and nVidia.