Origin of Space
How does something explode into a non existent space? It would seem to me that would be theoretically impossible!
In the Standard Big Bang theory, matter does not "explode into a non-existent space". Indeed, matter doesn't explode "into" anything. What happens is that the space BETWEEN things stretches. Here is a standard analogy: Consider a sheet of very stretchable rubber with little bugs on it. If you stretch the sheet in every direction by 50%, then every bug gets 50% farther away from every other bug, even if the bugs are not walking. Bugs that were an inch apart are now and inch and a half apart, while bugs that were a foot apart are now 18 inches apart. The bugs seem to have scattered in every direction, as if they were running away from each other. But it is not because the bugs are moving across the rubber sheet --- they could even be glued onto it. Rather it is because the rubber is stretching.
The space of the universe is like that, except that space has three dimensions and the sheet of rubber has only two. But you could imagine, instead of a sheet of rubber, a volume of stretchable rubber with lots of bugs stuck in it, with the volume getting stretched in every direction. (For the actual universe, do not think of "bugs" but of galaxies located at various places in stretchable space.)
In the analogy, the rubber sheet or volume presumably has an edge. But in the Big Bang theory, the space of the universe has no edge. There are two cases: (a) space goes on infinitely in every direction, which is called an "open universe", or (b) space curves around on itself, so there is only ever a finite amount of it, which is called a "closed universe".
One can easily imagine how a two-dimensional universe could be closed: just think of the rubber sheet being the surface of a balloon, which curves around to close on itself in a spherical shape. We cannot picture in our minds a three-dimensional space closing around on itself, because we cannot visualize enough dimensions. But it can be described using mathematics.
Suppose there are "bugs" distributed all over the surface of the balloon. Then, as the balloon stretches, the bugs get farther apart, but they are not expanding "into" previously previously empty places on the balloon. Similarly, one can imagine an infinite ("open") sheet of rubber, where every part has bugs on it. As the sheet gets stretched, every bug gets farther from every other bug, but they are not rushing "into" previously empty places.
If the universe is stretching, then as one looks back in time it was less and less stretched --- more shriveled, as it were --- the farther back one looks. If you look back far enough (about 14 billion years) space was completely shriveled up.
Wouldn't it be simpler to suppose that space itself is not expanding, but that the galaxies are just flying apart into a previously empty space? There are several problems with that. First of all, we know that in the early part of the universe's history (the first several billion years) the expansion was slowing (because of the mutual gravitational pull of all the matter in the universe. So the farther back in time, the faster the expansion was. Going back to very early times, the distance between galaxies (or the matter that would eventually become galaxies) was growing much faster than one light-year per year. If that was due to the matter moving THROUGH space, as you suggest, then the matter would be traveling much faster than the speed of light, which special relativity says is not possible. But if space was stretching, rather than the matter moving through space, there is no such problem.
Moreover, if one uses Einstein's equations of gravity (which have been tested in various ways and work very well), one finds that the matter in the universe causes space to expand just as in the standard Big Bang theory. Extrapolating those equations backwards in time, space shrivels to nothing about 14 billion years ago.
Nevertheless, if it is any comfort to you, it is possible that the Big Bang was not the beginning of space and time. It is also possible that in regions so far away that we will never see them even with telescopes space is empty. But for the part of the universe we can observe, the expansion is not due to matter rushing into emptiness, but due to space itself expanding as I have explained.