> At least, that's how I understand it. I could be wrong.

General Relativity deals in spacetime, but it is often convenient to slice spacetime into spacelike volumes in which every point in the volume ("the spacelike hypersurface") is at the same timelike coordinate. One important consideration is that no timelike axis is any more preferred by nature than any other, and one can always find an observer who disagrees with any choice of splitting one happens to make. In particular, two inertial observers related by a Lorentz boost generally will not agree on what event (e.g. two bits of matter colliding) is at what time coordinate, and thus this type of 3+1 slicing will result in different spacelike hypersurfaces for each such observer.

The cosmological frame picks out a frame that a family of special observers can agree on: these observers agree that at the largest scales, the universe is homogeneous and isotropic, that they themselves are moving inertially, and can agree on a function relating a time coordinate to the average density of matter in a universe-sized spacelike hypersurface at that time coordinate. Each such observer is assigned a spacelike location which persists into the infinite future: the observers are all stationary at a constant set of three spatial coordinates. The centres of mass of practically all galaxy clusters are essentially this type of observer, so those remain at the same spatial coordinates at all times too.

We then take this set of coordinates and apply it to a universe described by a Robertson-Walker metric. Our universe approximately obeys the Robertson-Walker (R-W) metric outside of galaxy clusters; more on that in a moment. The R-W metric relies on a 3+1 slicing of a homogeneous and isotropic universe, and uses two coefficients r and k to determine respectively the radius and shape of each spacelike 3-hypersurface. If we knock out a spacelike dimension, we can think of an R-W universe as a stack of infinitesimally thin plates where each plate at time t is related to its infinitesimally earlier predecessor and infinitesimally later successor plate by a function giving its radius r. (It is perfectly reasonable to rotate the axes so that you stack the plates vertically from the floor upwards, where we substitute a height coordinate for the timelike coordinate).

In an expanding-with-a-cosmological-constant R-W universe, r shrinks smoothly towards the past and grows smoothly towards the future. The coefficient k determines whether the 2-d planes give globally Euclidean geometry (k=0), globally hyperbolic geometry or globally spherical geometry. Finally, if r >>> the Hubble volume, there may be no practical way of recovering k != 0 observationally, or in other words the local geometry of a R-W slice of a Hubble volume may be flat even if the global geometry is not.

On this R-W universe we apply the coordinate system above, but remember that our observers stay at fixed spacelike coordinates. We need to notice here that our coordinates do not determine distances by light-travel-time. That's fine, we can use arbitrary measures of distance in General Relativity, and can practically always find a consistent and useful transformation from a description of physics in one system of coordinates to another. We just have to be careful either to work only in generally covariant formulations, or to recognize that using some systems of coordinates entice one into the use of fictitious forces that disappear in other systems of coordinates. In this case, an observer at the centre of mass of our galaxy in spherical coordinates centred on her would naturally say that distant galaxy clusters on this spacelike hypersurface now will be at a larger radial coordinate in future spacelike hypersurfaces, in our cosmological coordinates she and the distant galaxies are all working in coordinates comoving with r, so their spatial distance is constant at all times.

The metric expansion of space is just that: r increases.

> The inter-galactic gravitation doesn't pull hard enough to overcome the expansion

This gets trickier. In the Friedmann-Lemaître-Robertson-Walker model we treat the sources of the matter tensor as a set of perfect fluids with some pressure and density, and the fluids dilute away with expanding Robertson-Walker universe. We ignore the local overdensities of matter ("galaxies" and "people" and so on) and at the largest scales, that's reasonable.

However inside galactic clusters and galaxies, in the standard model there is no expansion at all; gravity doesn't work against it, it just isn't there in the first place. From a technical perspective what we do is treat galaxies as approximate sources of a Schwarzschild metric up to some boundary enclosing the galaxy, and then we embed that into Robertson-Walker space. This is certainly not faultless, but it matches observation extremely well. What does not match observation extremely well is naive quintessence models where the metric expansion works within galaxies and is simply checked by the gravitational interactions of the matter within them, but acts as a cosmological constant outside galaxy clusters.

Likewise, comoving galaxy clusters are just drifting along inertially into the future; even under a different system of coordinates there are no extra forces working to separate them -- throwing away the 3+1 slicing with its preferred timelike axis, in the spacetime view galaxy clusters' timelike worldlines converge near the hot dense phase of the universe.

If you took your second paragraph's metre stick[1] and put it in space as a comoving observer well outside galaxy clusters, it would still be a metre long in the far future, whether measured locally or with a really really really good telescope. "Rulers" aren't expanding in the metric expansion; instead the cosmological coordinates on each spacelike hypervolume are adjusted, and in general coordinates while useful are not themselves physical while an actual metre stick is. Physical objects themselves do not change when we change coordinates; distant galaxies can have no idea that you're putting cosmological or spherical or cartesian or conformal coordinates on them, or calculating their movements against those coordinates.

I'm afraid I don't understand the point in your second paragraph.

> two galaxies that are not moving with respect to each other

They aren't moving against comoving coordinates. But if you choose other coordinates (e.g. spherical coordinates with the origin on the centre of one of the galaxies) they can move against those. We can do various transformations to convert the descriptions of the motions of these galaxies (and any fictitious forces and relating to coordinate motion, and other coordinate-dependent quantities) in one set of coordinates into another set of coordinates. The trick is finding a set of coordinates in which one can extract some intuitions about observables like the cosmological redshift, the dark night sky (cf. Olbers's paradox), or the details of the cosmic microwave background.

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[1] in principle, and with some care, you could line up a hundred 1 cm objects (e.g. ball bearings) and they would not separate from each other with the expansion of the universe (one has to be careful about other things that may cause them to move relative to one another, such as radionuclide decay within the objects, interactions with cosmic rays or other particles, and so on; but in standard General Relativity they should continue on their parallel timelike worldlines indefinitely).