With computers you need to tell them everything you want to do. If you wanted to draw a picture you would need to tell each pixel to turn on in a way to make the picture you wanted.

That would take a lot of effort so someone made a program that would draw a line when you told it to draw a line. This moved into another level. Drawing a 3D environment where you could draw a ball by just telling this program to draw a ball. It would handle everything to make it look like a 3D ball on your screen with just a sphere equation. This was the foundation of a 3D engine.

Now these were used to make games, but some people went father and started making game engines where you can tell it to animate characters or do other game related tasks without telling the computer to do every step along the way. Things like real world physics were added so you could just tell the engine to make said ball and turn on, the ball would just drop and hit the ground and roll all without any extra programing required.

So a game engine is comprised of multiple different types of engines and programs that allow a game designer to focus more time and effort on designing games rather than other tasks.