Alexander Friedmann showed that if there is no extra cosmological constant, and the universe is filled with anything energetic (e.g. matter, radiation, dust, fluid, etc.) there were two classes of solutions: one for a contracting Universe and one for an expanding Universe.
It's as though the fabric of space itself is getting stretched over time, and all the objects within that space are being dragged apart from one another.
The farther away an object is from another, the more stretching occurs. If all of the universe was filled evenly with matter, that matter would simply get less dense.
Hubble discovery is of that individual stars could be measured in other galaxies, determining their distance.
On the largest scales, the expansion wins, on smaller - the own system gravity.
The superclusters of the universe, stretching for over a billion light years, are being stretched and pulled apart by the universe's expansion. Still, the Milky Way's nearest large galaxy cluster, the Virgo cluster, at just 50 million light years away, will not pull Milky Way into it.
Nearby, the Virgo cluster itself will remain bound. The Milky Way and all the local group galaxies stay together, eventually merging under their own gravity.
The fabric of space itself still expanding, but it doesn't have a measurable effect on every object/system.