Nicholas Copernicus, a Polish scientist living about a century before Galileo, had already come up with the unorthodox idea that the Sun was at the center of the solar system. Galileo knew about and had accepted Copernicus's heliocentric (Sun-centered) theory.
It was Galileo's observations of Venus that proved the theory. Using his telescope, Galileo found that Venus went through phases, just like our Moon. But, the nature of these phases could only be explained by Venus going around the Sun, not the Earth. Galileo concluded that Venus must travel around the Sun, passing at times behind and beyond it, rather than revolving directly around the Earth.
Galileo's observations of the phases of Venus virtually proved that the Earth was not the center of the universe. It was this assertion which most angered the Church leaders of the time.