What does 'space between spaces' mean?

I interpret Oxley's answer as "the aliens went to another dimension."

This makes sense as a follow-up to his earlier dialogue with Mutt Williams:

Williams: "What are they? Spacemen?"

Oxley: "Interdimensional beings in point of fact."

The aliens have come from inter-dimensional space. Wikipedia narrates:

Spalko believes that the crystal skull belongs to an interdimensional being and holds great psychic power, and proves her theory by showing that the corpse stolen from the warehouse has an identically shaped skull.

It further says:

After the Soviets arrive, Spalko replaces the skull. The aliens communicate to the group through Oxley using an ancient dialect and promises to reward them with a "big gift"; Spalko demands to "know everything". The skeletons grant her request and transfer their collective knowledge into her mind. At the same time they activate a portal to another dimension.

IMDB also confirms :

This refers to the fourth dimension. Definition in the American Heritage dictionary: "Time regarded as a coordinate dimension and required by relativity theory, along with three spatial dimensions, to specify completely the location of any event." There is a plausible theory in astrophysics that the fourth dimension may exist, which is utilized by the UFO in the film as the propellant engine to travel through the space/time continuum, as Nobel Prize winner in Physics Albert Einstein, who developed the theory of relativity, summarily explains: "An attempt at visualizing the fourth dimension: Take a point, stretch it into a line, curl it into a circle, twist it into a sphere, and punch through the sphere."

Oxley knew it. So he said Into the space between spaces.

Even though this was answered a while ago, here is more-or-less the same, but visualised.

Imagine two-dimensional beings. They live in 2D space, which we usually call a plane. There are infinitely many planes and some are parallel, i.e., they do not intersect. If you pick two of these, there is a space between those two planes (2D spaces), incomprehensible to 2D-beings:

Our beings here are 4D, so they can fathom the space between 3D spaces (for example, our and any other one parallel to it).

If we take the fourth dimension to be time, they can see and visit what you get between any two moments of our world (each moment being a 3D space, all of them parallel to each other), but I don't think that the movie went that deep.

Imagine 3 pieces of paper. As a 3d being, you can see all three pieces and write on any of them at will, then stack them up. Imagine a stick figure drawn on one of the outer pages (the equivalent of our world). You (the aliens) put the piece where the stick figure is to the side and begin drawing on the middle page (the space between spaces). The stick figure could never possibly comprehend that there is any such thing as another piece of paper, but they can still see you. You of course seem to them as a superior being, and if you tried to show them the other piece of paper, they would be entrenched with a mass amount of information and vaporize just as Spalko did. Hope this view can kind of put it in perspective for you.

I know this is an old topic but I really like this question. Here are my two cents.

In Vedantic hinduism, Akasha means "the basis and essence of all things in the material world; the first material element created from the astral world." In vendatic philosophy it means "an ethereal fluid imagined as pervading the cosmos"

Another synonym for akasha is æther, ether, or quintessence, and most notably the fifth element. In Greek philosophy these mean "the material that fills the region of the universe above the terrestrial sphere. " aka the space between spaces. These are not my beliefs, aside from my perception of the movie quote, but they are historical facts. Sorry for poor grammar and writing.