Galileo saw the four largest moons of Jupiter in orbit around
the planet, proving that the Ptolemaic system was not simpler - the
solar system was not geocentric (the planets and the sun did not
orbit around the earth).
As far as the observations of Venus go, Galileo saw that Venus displayed phases very much like our moon. Now, according to the Ptolemaic system, Venus could only display a crescent phase because its epicycle put it always in between Earth and the sun. The Copernican system put everything rotating around the sun, and in this way it explained the phases of Venus.
Think of it this way, if Venus is always in between the sun and the earth, how can we ever see it completely lit up, like a full moon? The sun is always on the other side of it, so Venus must at some point go on the other side of the sun.