If spacetime is expanding, shouldn't lengths expand at an equal rate? I frequently hear that galaxies are moving apart (in some instances, faster than the speed of light) because the fabric of spacetime is expanding.  The common analogy is two people standing on the surface of an expanding balloon.
However, this makes no sense.  Since we are made of matter that exists within spacetime, shouldn't we (and all of our unit lengths) be expanding at the same rate?  In the balloon analogy, if the people are drawn onto the balloon, they will expand at the same rate as its surface. If you draw a coordinate grid onto the balloon, distances measured using those coordinates will stay constant as the balloon expands.
So what's really going on?

Note: Astrophysicist Nick Lucid from Science Asylum explained in a Youtube video that it's because electromagnetic force holding our atoms together is so strong, it keeps us at the same size while spacetime expands. But that explanation makes no sense because it would mean the electromagnetic force gets stronger over time (drops off by less than $\frac{1}{r^2}$), and also doesn't explain why galaxies would be spreading apart.

[Edit] PBS Spacetime just released a video about this exact question, which does a better job of answering the question than anything else I've seen.
 A: 
If you draw a coordinate grid onto the balloon, distances measured using those coordinates will stay constant as the balloon expands.

You are correct if we talk about the FRW model with perfect fluid where the matter density is constant regardless of the scale.
But the inhomogeneities in our actual universe as there are star systems, galaxies and galaxy clusters change the picture. These systems are gravitationally bound such that they resist the expansion on the universe. In other words they don't participate in the expansion of the universe. Coming back to the ballon analogy it is as if such systems were connected by invisible threads. Only even larger structures e.g. super clusters are expanding so that the proper distances between the clusters are increasing.
Note that the coordinate grid is a mathematical tool. What matters are not coordinates but the energy densities according the FRW model which determine the evolution of the universe over time.
A: Well, if we were to expand relative to the same rate spacetime is expanding, it would be the same like saying space isn't expanding at all. Since we would grow as spacetime is growing around us, the distances between galaxies would remain the same (if everything grows in proportion to one another) and we would not see distant stars and galaxies moving away from us.
And astronomers can see galaxies moving through space, because we can detect the radiation emitted by these distant stars and galaxies and see a shift in wavelength by getting longer and more stretched out. If everything within spacetime where to expand in the same rate, we would not see an increase in wavelengths from distant galaxies.
Also we seem to forget about good old gravity, curvature of spacetime is what keeps our solar system in orbit around the sun. Other galaxies have their own mechanism of what keeps them in orbit. Spacetime is expanding but we are bound together within our solar system just like other planets are orbiting other stars. While everything within spacetime is moving at the speed of light, we could also say everything remains stationary and spacetime around is expanding, pushing galaxies apart.
