I need a very detailed description of how the coffee machine with programmable timers actually work. I need enough detail and instruction that I could build one. Links to images of the inner workings of a coffee maker would be great too, but not necessary. Best answer will go to the instructions that I could build a programmable coffee maker from!

It is simply a switch built controlled by a timer that’s all.
It uses a programmable clock/timer chip, usually one custom made for the machine. In some older ones they used a digital clock radio chip.
The control chip, whether custom or off-the-shelf, would basically be a specific logic circuit, or programemd mictocontroller, interfaced to an LCD or LED display, a few buttons, the AC line for a time base, and and a line to control the brewer, through a relay.
On the old ones, they were dumb, in that they turned the whole coffee maker on or off. Brewing was complete when the water ran out, and a rise in temperatur if the now dry water boiling chamber tripped a thermosat to turn it off.