The concept of the expansion of the universe can be a bit counterintuitive, so let's break it down. When we say that the universe is expanding, it means that the distances between galaxies (on average) are getting larger over time. However, it's important to note that the expansion of space itself does not cause galaxies to physically move away from each other like objects moving apart on a surface.
The expansion of the universe is described by a concept called the Hubble's law, which states that galaxies that are sufficiently far apart from each other exhibit a redshift in their light. This redshift is a result of the stretching of space itself as the light travels through it. The farther away a galaxy is from us, the more the space between us and the galaxy has expanded, causing the light from that galaxy to be stretched to longer wavelengths, resulting in a redshift.
Now, it's true that nearby galaxies, which are gravitationally bound to each other within galaxy clusters, do not exhibit this redshift due to the local gravitational forces overpowering the expansion of the universe. So, it is not accurate to say that galaxies are moving away from each other faster than the current expansion rate on small scales.
However, on larger scales and when we look at more distant galaxies, we do observe the redshift and the effects of the expanding universe. In fact, the observations of the cosmic microwave background radiation, which is the remnant radiation from the early universe, provide strong evidence for the expansion of space.
In summary, the expansion of the universe refers to the stretching of space itself, which leads to an overall increase in the distances between galaxies. While nearby galaxies may be held together by gravitational forces, on larger scales, we do observe the effects of the expanding universe through the redshift of light from distant galaxies.