How do geocentric theories explain seasons? Did the geocentric theorists also use the fact that the earth had an axis tilt? Or was their explanation of some other kind?
 A: The obliquity of the ecliptic and the solar eccentric (a slight shift from the center of the Earth of the center of the Sun's path around the Earth).
Imagine the "fixed stars" as being points of light fixed to a bowl that rotates around the Earth on a daily basis. That's how the ancients thought of the stars. Plot the position of the Sun at noon every day with respect to this bowl of stars you'll see a pattern arise: The Sun moves with respect to the fixed stars. This is the ecliptic. The planets also appear to more or less follow the ecliptic. The ancients knew this very well. It was a part of many of their religions.
The way this was explained from an ancient geocentric point of view is that the rings on which the Sun and planets rotated moved north and south, making the Sun and planets appear to move along the ecliptic. This nicely explains the longer days in summer, short days in winter.
What it doesn't explain is why spring and summer are longer than are fall and winter in the northern hemisphere. This too was well known to the ancients. Explaining this phenomenon is where the solar eccentric comes into play. The the Sun and planets moved in circles called deferents. (The planets needed epicycles, too.) The center of a deferent, the eccentric, is slightly offset from the center of the Earth in Ptolemy's model. The Sun appears to move uniformly from the perspective of the deferent, which means it doesn't appear to move uniformly from the perspective of the Earth. Ptolemy explicitly took the non-uniform lengths of the seasons into account when he established the Sun's deferent and eccentric.
