From another question I posted, I learned that not only was matter created from the Big Bang, but more importantly, space itself. What does it mean for space to be created? Does it exist in the same way substances exist? In other words, what do we mean by 'space exists?'
This is a very complex question, and perhaps more appropriate for the Physics or Philosophy stacks. I will attempt to give you an initial direction, but really I think this is too broad for this stack. I'm merely writing here because comments are too short.
In a very practical way, space is simply that which allows objects and events to have relative position, direction, and extent. There is still disagreement over whether space is a thing in itself, or merely a relationship between things, or part of a conceptual framework. See wiki
In relativity, the dominant concept is not space, but spacetime - closely related to a fundamental constant of the universe (speed of light) and to gravity. Space itself, or time itself, are less important, but the 4-dimensional metric of spacetime is far more fundamental, in a sense.
Does it exist in the same way substances exist?
We don't know. See the wiki link for the philosophical debate.
what do we mean by 'space exists?'
It means this universe is built in such a way as to allow relationships of position or direction. This is a bit of a circular definition, but it's the best we have.
What does it mean for space to be created?
Actually, that happens all the time, even now. The so-called dark energy actually makes more space, continuously, even now. That simply means that, whereas a while ago there was X amount of distance between galaxies A and B, now there's X + Y amount of distance. The low-level mechanism that allows this creation is not well understood.
It could be imagined that something similar happened during the Big Bang, except it started from zero. Again, the low-level mechanisms around the origin are not well understood. To understand that we first must be capable of understanding something far more simple - the physics of the center of a black hole. To do that, we would need something like quantum gravity, which is a part of Physics that has not been invented / built yet.
If this seems overly speculative, it's because you're asking a very profound and difficult question - so difficult that it exceeds the limits of current science and even philosophy.
Space is considered to be the volume in which stuff (eg. stars) exists. It is also considered that it can expand everywhere, resulting in more space between the stars, without the stars having moved away from each other.
The usual analogy is the expanding balloon, with dots representing stars or galaxies, and the surface of the balloon representing empty space.
As you blow up the balloon, its surface area expands (ie. like space expands), and the space between the dots (stars) increases. If you were living on a planet around one of the stars (dots on the balloon), you would observer all the other dots/stars appear to move away from you, and if you were on any of the other dots, you would observer the same thing.