The expansion of the universe is a separate issue from the internal motions of galaxies. They are more like large solar systems, which are not themselves expanding, but are moving away from other galaxies (actually other clusters of galaxies). For example, the solar system is not expanding - the earth is not moving farther away from the sun. Similarly the solar system is not moving farther away from the center of the Milky Way.
For each galaxy, there is a center of the galaxy, just like the center of the solar system is the Sun. Very interesting if all galaxies at their boundaries have the same rotational speed. However, the little movie is not correct. A galaxy is not a solid body rotator, like a record or CD. Look at the article from Wiki on the rotation curves of galaxies. If the galaxy rotated as a solid body, then the rotation curve, which measures the speed vs. distance from the center, would be a straight line. At the distance of the solar system, the galaxy takes about 250 million years to orbit around its center. For a long time it was assumed that most of the mass of the galaxy was concentrated at or near the center of the galaxy (that is the implication of the density of starlight toward the center.) However, it was determined, by measuring the speed of the stars and the gas between the stars that there must be a huge amount of material within galaxies that cannot be accounted for by the stars or gas. This was because, instead of the speed of stars and gas farther away from the center slowing down, like planets farther from the sun, they are actually speeding up. This unseen matter is called dark matter, and has been calculated to make up most of the mass of a galaxy. Given their rotation curves, galaxies would fly apart without this dark matter holding them together. The fact that we can date the age of galaxies by the age of their oldest stars, means they have been stable for many billions of years and so must have something holding them together.
https://en.wikipedia.org/wiki/Galaxy_rotation_curve
Larry, your astronomer friend