Copernicus’ heliocentric model of the cosmos stood in opposition to the spheres model, which posited that the motion of the planets was caused by the circular motion of celestial spheres, surrounding the earth, in which the planets and stars were embedded. The idea was that each sphere drove the motion of the sphere within it it, which conveniently made the need for an unmoved mover (equated with God in a Christian context) to drive the motion of the cosmos.
Galileo’s role in all of this was his publication of Sidereus Nuncius, which detailed his observations with the telescope. One observation he made (which we take for granted today), was that the moon had surface features (craters, mountains, etc.). This contradicted the Aristotelian assertion that the heavenly bodies were smooth spheres, being more perfect than Earth. Worse still were his observations of four objects which traveled in a straight line near Jupiter, which Galileo correctly concluded were orbiting Jupiter, an impossibility in the celestial sphere model, since objects should only orbit the Earth. This basically was an indication that the Aristotelian model had some huge problems with it.
The final piece of the puzzle was Kepler’s law of planetary motion, which posits that the planets move in elliptical orbits. This, combined with Copernicus’ heliocentric model, created a much simpler and more accurate system to calculate the movement of the planets, as opposed to the Ptolemeic system of epicycles and deferents, and which when combined with Newtonian mechanics and the Law of Universal Gravitation predicted to just one degree of error the location of the formerly unknown planet Neptune, based on periodic deviations in the orbit of Uranus.
With all of this evidence, I think we can say pretty conclusively that Copernicus’ heliocentric model, combined with Kepler’s Law of Planetary Motion, and Newton’s Law of Universal Gravitation is a much better explanation for planetary motion than the Aristotelian geocentric model with celestial spheres.