What is the meaning of the "expansion of space"? When we say that "the space between galaxies is expanding," what do we really mean? For instance, if I think of space as being a Cartesian grid, then when space expands should I think of it as adding more grid-points or as making the distance between the grid-points larger? Or is this a flawed picture of space-time even when far from significant mass densities?
It seems to me that it must be the case that we are "adding more grid-points" because otherwise we would not observe the expansion since anything that occupied that part of the grid (e.g. light, my hand, etc.) would also expand by a corresponding amount (otherwise, after a sufficient amount of expansion, we would be able to observe sub-atomic scale processes as being macroscopic in scale). But if it is the case that we are adding more grid-points, then the "expansion" of space seems like a misnomer: shouldn't it be called the "creation" of space?
That being said, either case (adding grid-points or stretching distances) could conceivably be the same process depending on the nature of space-time (i.e. if it were discrete but dense - like the Rationals - rather than either strictly discrete (Integers) or strictly continuous (Reals)... I hope that analogy makes sense). 
So what is really going on in intergalactic space? What is the meaning of its expansion? Does it correspond to creating more space, stretching the space that's already there, or something more subtle?
 A: I could start this answer by saying that space isn't really the thing that's expanding; that it's the scale factor of the metric that's growing larger. But I won't because that's just an overcomplicated way of saying space is expanding. But to answer your question, the most correct way to think about it would be that the grid points are getting farther apart. It is not that we are adding more grid points.
You are correct that the "stretching" of space would apply to all things. For instance, we know that as light travels through space, the expansion causes its wavelength to expand correspondingly. The reason we do not expand with it though is that we have forces holding us together. The space between our atoms does not expand because the electromagnetic force is stronger and holds our atoms in relatively the same place. Similarly, the Sun's gravity holds Earth in its orbit in spite of the expansion. In fact, for all objects within our local cluster, gravity is strong enough to counteract the effect of the grid points separating.
TLDR: The expansion of space is more like "stretching" than adding new space
