Consider a situation where everything is removed from the Universe. All matter, all radiation, everything.
Only a completely black void would remain. But what is it? What is this empty "space" that is nothing? Or would this nothing even exist without matter in the first place?
It is very hard to try to explain or imagine this.
Edit: What I'm trying to ask is that what does it mean that there's room for particles to move? By "space" I don't mean the Universe, but space.
As with much in science, it is best to talk in terms of "models" rather than what something is. We can model a photon, for example, as a quantum excitation of the electromagnetic field.
Space is part of the model for doing physics. Together with time it is a coordinate system to describe locations, and the distances between locations.
The simplest model is "Euclidian" in which the positions are described by triplets of numbers $(x, y, z)$ and distance is $\sqrt{x^2+y^2+z^2}$.
This is especially important in General Relativity as the "curvature of spacetime" is defined in terms of formulae for describing the distance between points (a "metric"). In relativity spacetime is $(x,y,z,t)$, and the distance has a more complex defintion.
You can conceive of empty space, and it is useful in some situations. In particular there are important metrics that apply to empty space: The Minkowski metric of flat space, and the Schwatzchild metric of empty space around a point mass.
So what is space(time), is a question for philosophers. How do we describe, use and model spacetime is a practical question with a specific answer. We can describe and model space using a coordinate system.
There can be no such thing as "empty" space. Particles and anti-particles can and do pop into and out of existence on timescales that are shorter than $\sim \hbar/\Delta E$, where $\Delta E$ is the total energy (including rest mass) of the created particle/anti-particle pair.
Thus your thought experiment is impossible.
This is a philosophical rather than a scientific question. Space and time are not physical objects but forms of our existence. Without space no objects could exist, and without time no change of the configuration of physical objects (i.e. no motion) would be possible. You are effectively asking what the world would look like if there wasn't any world. That is a logical contradiction in terms.
Physicists refer to space with no matter or energy (radiation) in it as the vacuum. You'd think that since it contains nothing, it would have no physical properties- but it does.
The vacuum propagates electric and magnetic fields in precisely measurable ways even though it contains no matter. For example, capacitors work even with vacuum between their plates which means space has a certain dielectric constant that is nonzero, even if its matter content is zero.
The study of the properties of the vacuum is an active field of physics research in which steady progress has been made over the years. Before quantum mechanics was discovered, what was known about the vacuum was summarized in a model called the classical vacuum, which was later replaced by the vacuum containing quantum fields. Check out wikipedia for more background on these topics.
