You can think of them as functions that represent "how the rest of the computation proposes to continue." Except that, once we're able to get our hands on those functions, we can do exotic and unwholesome things with them. Like use them to suspend and resume a thread. Or to abort from deep inside a sub-computation: one function might pass the command to abort *it* to a subfunction, so that the subfunction has the power to jump directly to the outside caller. Or a function might *return* its continuation function to the outside caller, giving *the outside caller* the ability to "abort" the function (the function that has already returned its value---so what should happen then?) Or we may call the same continuation function *multiple times* (what should happen then?). All of these weird and wonderful possibilities await us.
You can think of them as functions that represent "how the rest of the computation proposes to continue." Except that, once we're able to get our hands on those functions, we can do exotic and unwholesome things with them. Like use them to suspend and resume a thread. Or to abort from deep inside a sub-computation: one function might pass the command to abort *it* to a subfunction, so that the subfunction has the power to jump directly to the outside caller. Or a function might *return* its continuation function to the outside caller, giving *the outside caller* the ability to "abort" the function (the function that has already returned its value---so what should happen then?) Or we may call the same continuation function *multiple times* (what should happen then?). All of these weird and wonderful possibilities await us.