High speed communications on a lower speed controler
Such fast outputs are never handled directly by a processor. That would be called "bit banging" and is really only practical for bit rates that are a small fraction of the CPU instruction cycle rate (note that a chip with an 8MHz external clock might internally run at 2MHz or 200MHz instruction rate).

Instead there are peripherals that are dedicated arrangements of gates, flip-flops and registers that are used to serialize data, perhaps directly from memory (DMA = direct memory access). There may be an internal memory bus dedicated just to getting data out of a frame buffer at that rate. In the case of HDMI there would likely be a dedicated graphic processor (GPU) optimized for video processing. 

In the case of the Broadcom BCM2835 SOC used in the Raspberry PI, the GPU is a VideoCore 4 ARM1176. It would be nice to see the internals of such a chip, but Broadcom holds the cards close to the chest, and an NDA and a substantiated intent to use a substantial volume of chips is the price of entry.