> But inside the model there really isn't a bijection between the sets of naturals and reals
No. Inside the model, there is exactly such a surjection. We don't care what happens outside of it.
Maybe you already knew this, but to clarify: "Countable" means that there exists a surjection, not necessarily a bijection. For instance, the set {a,b,c} is countable, but there is clearly no bijection Nat -> {a,b,c} as one set is infinite and the other's finite.
> It's not too surprising
On the contrary: It's a ground-breaking result IMO.