The way I interpret is that it is possible to find an algorithm to compress a set of data points in a way that is also good at predicting other data points, not yet observed. In yet other words, a good approximation is, for some reason, sometimes also a good extrapolation.

Well, yes, and the reason isn't mysterious.

In order to compress a stream of data you need to discover some structure in it. If there is no structure -- e.g. if the stream is truly random -- then no compression is possible. And if the structure you found is "really there" and not an artifact of your structure-searching techniques, then it just as useful for extrapolation and prediction.