I could understand an initial explosion imparting momentum to matter but where is the energy stored in space itself? How does empty space have momentum that causes this expansion?
We can't exactly say what caused the universe to expand, because we don't know its origin. The Big Bang was not an explosion, at least not as far as we know: it was simply a moment in the past when the universe had a very high density. All we can really say is that according to the laws of physics (specifically, general relativity), the presence of matter in the universe makes it so it has to either expand or contract. Why does it expand instead of shrinking? We don't know - we just know that at some point in the past it was expanding, so it keeps expanding now.
But to answer the question in the text, yes, spacetime can have momentum - sort of. It would be more accurate to say that it has dynamics: the curvature of spacetime is affected by the presence of matter and energy, but it has its own behavior, and it obeys its own set of laws. There's no energy or momentum anywhere; what there is is curvature, which changes and evolves in time and space.