Take a min to think about it. I think we can agree that the big bang started in a infinitesimally small point and is the source of all matter, and before the big bang, there was absolutely nothing. So that will be where I'll base my points.
Thus based on this definition, before the big bang, there was nothing. Space is something, empty space now contains quarks, photons, electrons, neutrons, etc... flying around randomly, but before the big bang, there was nothing. Yes, there kinda was time there but how do you tell the difference between one moment to the next when there is nothing at all there, and why would it matter? So before the big bang, time kinda didn't exist is the notion.
Since the big bang itself is the source of all matter, it could be imagined as a bubble that our universe just started as a point and started growing with all the matter and bits and pieces already inside it since it was a point. Since that bubble with all the stuff in it never existed until the big bang, his phrase "'here' (space) didn't exist yet" is technically true.