Here's my (probably weak) understanding of how it works: when a continuation is captured a chain of exceptions gets propagated up the stack. At each "stop" on the way up the stack, an exception handler cons-es up a closure which will continue the function whose frame is active at that point. When the exception reaches the top level of the stack, the frames are packaged up as the continuation. Activating the continuation simply calls the closures in order to resume the saved computation where it left off. All the implementation needs to provide is some way to unwind the stack, stopping at each frame along the way. In C, it would be a longjmp/setjmp; in Java it would be catch/throw; in .NET catch/throw.
There is an example with more code (and less math) here, too.