The key idea behind working with continuations is that we're *inverting control*. In the fragment above, the code `(if x = 1 then ... else outer_snapshot 20) + 100`---which is written as if it were to supply a value to the outside context that we snapshotted---itself *makes non-trivial use of* that snapshot. So it has to be able to refer to that snapshot; the snapshot has to somehow be available to our inside-the-box code as an *argument* or bound variable. That is: the code that is *written* like it's supplying an argument to the outside context is instead *getting that context as its own argument*. He who is written as value-supplying slave is instead become the outer context's master.
The key idea behind working with continuations is that we're *inverting control*. In the fragment above, the code `(if x = 1 then ... else outer_snapshot 20) + 100`---which is written as if it were to supply a value to the outside context that we snapshotted---itself *makes non-trivial use of* that snapshot. So it has to be able to refer to that snapshot; the snapshot has to somehow be available to our inside-the-box code as an *argument* or bound variable. That is: the code that is *written* like it's supplying an argument to the outside context is instead *getting that context as its own argument*. He who is written as value-supplying slave is instead become the outer context's master.