The cosmological principle says that the universe has roughly the same average density of matter everywhere. According to Einstein's general theory of relativity, gravity can be regarded as a curvature of space and time. Matter causes gravity.
Gravity makes matter move.
In 1917, Einstein applied his theory of general relativity to the entire universe. The theory predicted that the universe must be in motion. In 1920, Alexandre Friedmann solved Einstein's equation without the cosmological constant. The solutions indicated that the universe started as an explosion. It will either expand forever or collapse again.