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It is not like a continuation, it is exactly a (typed) continuation, or better, an infinite list of continuations (invoking the continuation.
In your example of reading an int the EffectHandler and Effect are simply switched (better names are sink and source). And yes, for IO you will need a scheduler, but streams are much more straightforward.
I have reached my limit of pure functional language knowledge, but I can offer you a working implementation [1] in an imperative language.
I've actually implemented these typed continuations in c++ years ago, and I'm trying to understand how they differ from effects (aside for the whole imperative thing).
In the C++ implementation, for convenience the continuation object is replaced with the next continuation when invoked, but internally actually invoking a continuation function returns the yielded value and the next continuation as for my EffectHandeler example.
https://github.com/gpderetta/libtask/blob/master/tests/conti...