Why the curve stays flat
A galaxy rotation curve plots the orbital speed of stars and gas against their distance from the galactic center. Based on the visible mass, Keplerian dynamics predict speeds should fall off with distance, the way outer planets orbit the Sun more slowly. Observations instead show the curve flattening: stars far from the center orbit roughly as fast as those nearer in.
Vera Rubin and Kent Ford documented this for many spiral galaxies in the 1970s, building on earlier hints. The flat curve implies far more mass at large radii than the luminous matter accounts for.
The dark matter halo interpretation
The leading explanation is an extended halo of dark matter whose gravity holds the fast outer stars in orbit. Rotation curves remain one of the cleanest pieces of evidence for dark matter, alongside gravitational lensing and the cosmic microwave background. Alternative theories such as Modified Newtonian Dynamics attempt to explain the curves without unseen mass but face challenges at cluster scales.