I'm very far from being an expert, but as far as I understood the reasoning, it seems that the author is arguing that as time passes, the current conditions of

I'm very far from being an expert, but as far as I understood the reasoning, it seems that the author is arguing that as time passes, the current conditions of chaotic physical systems are going to start depending on more and more information regarding (or "binary digits" of) the variables defining the initial conditions. Since the deterministic equations of classical physics are defined over the real numbers but physical variables cannot contain infinite information, at some point some of the information/binary digits of the initial conditions that (partially) defines current conditions must be a random digit -- although this conclusion is only reached after arguing that almost all of the real numbers are numbers that contain random digits (as random as they can be, because they have no structure and can't be named/characterized/defined). As a layman, I'm not really convinced by this reasoning (if indeed this is what the author argues), but this is the gist of what I understood.