Okay, so.

Take a balloon, uninflated, and take a marker and put a bunch of dots onto the outside of the balloon.

Each dot represents a galaxy. Put as many dots as you can onto the balloon.

Now, blow it up.

Notice how the dots are spreading away from each other even though they're not moving?

That's the universe. The universe itself is expanding like a balloon, and the galaxies are moving apart from each other.

The galaxies stay together as galaxies due to something to do with dark matter that we don't yet understand.