> at a certain rate
No, the evidence suggests that the rate is time-dependent; that is, the expansion is accelerating.
> how do all the parts in space coordinate to expand at about the same steady rate?
This is a good question!
The answer is: the coordination happens in the much earlier much denser universe in our past, just like the coordination that happened among the fields of the Standard Model that led to the earliest atoms and molecules.
Let's first take the case where there is no accelerated expansion (which is unlikely with current evidence). In this case, the mechanism can simply be inertia, with an enormous early impulse like those in the family of cosmic inflation theories. There must be a fine balance between the density of matter in the earlier universe and the impulse, because gravitational interactions will decelerate the expansion from the early impulse until the matter-density drops/dilutes-with-expansion to a critical point after which the rate of expansion will be constant.
Observations support an early deceleration and subsequent fixed-rate expansion, but about 8 billion years ago the expansion began to accelerate, as seen in images of galaxies with redshifts less than about z ~ 0.5.
> a simple underlying mechanism
This is what the study of dark energy is about: a simple explanation for the accelerating expansion of the universe.
Typically one starts with a field that fills the whole of spacetime, with enormous energy-density in the early universe, with that energy-density decaying at later times. In extremely high-quality vacuum and far from structures like stars and black holes, the decay of this field is faster than near those structures (too near and the field becomes locally stable; the solar system, for example, is not expanding). When the field decays, it gives a local increase to the inertial expansion rate described in the no-dark-energy case a couple paragraphs up. Because there is so much empty space outside galaxy clusters, there is a lot of local, and the amount of local increases over time, so larger regions of the field decay at faster rates as the unvierse gets older.
There are other approaches too, but the effect is the same: the amount of space outside matter structures increases at an accelerating rate in the late universe, and this corresponds in a frame-dependent way to an energy-density that remains constant as the energy-density of the fields of the Standard Model decreases. That is, in such a frame of reference, matter and radiation dilute away with the expansion MUCH faster than dark energy does.
Another approach is literally the cosmological constant, which if zero gives the inertial picture above, but if it takes on a small positive value, it produces a late-time acceleration of the expansion. No known fields behave like this: matter-density varies by location in spacetime (clumpier in space, tending to diffuse apart (especially light) or collapse gravitationally (especially baryons) over time), gravitational fields vary with position in spacetime and typically with the distribution of matter. By contrast, the cosmological constant is literally identically everywhere-and-everywhen. More likely we will find evidence for something which closely approximates the cosmological constant at large scales, with differences in the density of the "cosmological not-quite-constant field" lining up very closely with observed differences in the density of matter.
> clever nature of this mechanism
Cleverness shouldn't be expected of the mechanism itself, although it will probably take some cleverness to describe the mechanism in a tractable/scalable-yet-accurate-in-detail manner.
WRT your final sentence, General Relativity works just fine in the total absence of dark energy, and dark energy as the cosmological constant has been captured as the factor \Lambda in the Einstein Field Equations. That's like saying that all matter has been captured as a factor in the Einstein Field Equations, though: it has taken a lot of work including things like studying https://en.wikipedia.org/wiki/Deep_inelastic_scattering to work out many of the finer details of the stress-energy tensor. A comparable program likely is needed to work out the finer details of \Lambda, which might lead to a formal explanation along the lines of the informal one a few paragraphs above.