Earlier quoted context omitted.
It seems to me that the fundamental thing that Wolfram is saying is that reality is non-local. Locality itself is an emergent phenomenon. This is his clever way of getting around Bell's Theorem. His networks don't require space to exist, they are space. When we think of spacetime, we think of a manifold, and he's saying that what looks like a manifold at a macro level is really just a discrete network at a micro leve…
But a theory is supposed to model something that actually exists, yes? So for instance when philosophers made the conjecture for the existence of the atom, their theory was either correct or not, on the basis of whether the atom actually existed in nature. So if Wolfram is correct, then there exists at the lowest level of nature a network, upon which everything else follows. Wolfram is trying to describe reality, som…
Over time, physicists have come to take models very seriously (thanks in part to Einstein's 1905 papers). That is, when our model says that electrons act like particles and waves, we assume that electrons really are these odd things that are somewhere in between a particle and a wave.
So, if it turns out that Wolfram's ideas have merit, and there really is an underlying spacetime network which can be used to model and derive all of the known equations of physics (and likely some unknown equations), then at that point we would have to take very seriously the existence of such a network. We would be forced to think of a seemingly continuous spacetime as a special facet or feature of this underlying network, in the same way that we now think of space as a special feature of the more general spacetime.