Dr. Bee explains it in her weekly podcast:
If the universe expands the same everywhere, then doesn’t this define a frame of absolute rest. Think back of that elastic band again. If you sit on one of the buttons, then you move “with the expansion of the universe” in some sense. It seems fair to say that this would correspond to zero velocity. But didn’t Einstein say that velocities are relative, and that you’re not supposed to talk about absolute velocities. I mean, that’s why it’s called “relativity” right? Well, yes and no.Relativity teaches that there is a symmetry transformation from one inertial frame to another. The laws of electromagnetism and other physics are transformed by covariance. But there can still be something that distinguishes a frame as being at rest, and that something is the cosmic microwave background radiation.
If you remember, Einstein really had two theories, first special relativity and then general relativity. Special relativity is the theory in which there is no such thing as absolute rest and you can only talk about relative velocities. But this theory does not contain gravity, which Einstein described as the curvature of space and time. If you want to describe gravity and the expansion of the universe, then you need to use general relativity.
In general relativity, matter, or all kinds of energy really, affect the geometry of space and time. And so, in the presence of matter the universe indeed gets a preferred direction of expansion. And you can be in rest with the universe. This state of rest is usually called the “co-moving frame”, so that’s the reference frame that moves with the universe. This doesn’t disagree with Einstein at all.
What is the co-moving frame of the universe? It’s normally assumed to be the same as the rest frame of the cosmic microwave background, or at least very similar to it. So what you can do is you measure the radiation of the cosmic microwave background that is coming at us from all directions. If we were in rest with the cosmic microwave background, the energy in that radiation should be the same in all directions. This isn’t the case though, instead we see that the radiation has somewhat more energy in one particular direction and less energy in the exact opposite direction. This can be attributed to our motion through the restframe of the universe.
How fast do we move? Well, we move in many ways, because the earth is spinning and orbiting around the sun which is orbiting around the center of the milky way. So really our direction constantly changes. But the Milky Way itself moves at about 630 kilometers per second relative to the cosmic microwave background. That’s about a million miles per hour. Where are we going? We’re moving towards something called “the great attractor” and no one has any idea what that is or why we’re going there.