How does a game engine/graphics engine work (software)?

well in a sense they are like engines.
a classical engine takes petrol and oxygen and converts it to forward motion.
a game engine takes user input, premade models, animation sequences etc and converts it into simulated events and a successive stream of rendered images.
the differences between game engines is how they do that, what and how many steps they take to produce the images and events.
In that sense they are again very similar to real engines, which vary widely in design and performance.

Put simply:

The unit itself (PC, PS3, Xbox 360 hardware, etc) is a machine, capable of a myriad of things. It has millions parts, doodads, levers, buttons and switches, all available to the developer. However, certain combinations work better than others, and if you switch them all on at the same time, the machine won’t run well (low framerate, clipping, pop-ins, etc.)

The engine (CryEngine, Unreal, etc) is like a set of directions that tell the machine how to operate. Middleware (as it’s called) is sold out to developers so that they can use it as a platform to build their own game. The better the engine performs (and easier it is to use), the more popular it is. Of course, these engines are easily modifiable, so a back-end coder can change stuff around to their liking, if need be.

The game developers just go in there, add their character models, environments, animations, lighting, textures, music, voice acting, etc.

That’s the simplest I can put it.

It is called and “engine” because it is the part of the software that does most of the computations; so it “drives” the rest of the programming.

When you play a game, like Doom, where you seem to move around in some room or dungeon or whatever, every time you move, the illusion of movement is created by the walls and other objects moving past you. That movement of, say, a picture on the wall moving past on your right, is done through calculating how fast you are moving and in what direction. Those calculations are used to move the appearance of that picture so it appears to move toward you.

The “engine” is a set of computer code that does all the calculations. It has certain inputs and produces specific outputs for the game. In this way the people who are designing the appearance of the dungeon don’t have to do all the coding for all the calculations, they just tell the engine what is happening and it does the calculations for them.