Seasons are caused by the axial tilt of the Earth. Fairly good estimates of the axial tilt were made in antiquity, and better ones were made during the Early Modern era. If you are the sort of thinker who cares about things like axial tilts, you are the sort of thinker who is going to get that this affects things like seasons.

Why would they measure and know this kind of thing in antiquity? Because seasons are important if you are dependent on agriculture. Hence most agricultural societies have complicated calendars, ways of measuring (and festivals associated with) the solstices, and other ways of physically marking the passage of time so that they know when it is time to plant, harvest, etc. Tracking the movement of the Sun over the course of the year, and correlating that with seasons, is a very very old endeavor. (And it works whether you assume a geocentric or heliocentric universe.)

As for proving it — this isn't really the right question to ask, I don't think. The interest in proving theories (over competing ones) assumes a "scientific" mindset. This is observational astronomy coupled with agricultural requirements — one need not "prove it" if it gets the right results, and the impulse to prove things through induction and/or deduction is a later, separate sort of mindset.

Anyway, this doesn't totally answer your question — I suspect it is impossible to put one's finger on the exact way in which humans "learned" this, because it is such an old approach to the sky. The notion that all ancient peoples thought that seasons and stars and etc. were purely supernatural is a mistaken one, even if their understandings were not correct or precisely scientific by modern definitions. There were great practical stakes associated with understanding these things on a functional level, even if sometimes layered upon that level was a more mystical explanation. (Arguably some of the scientific explanations we layer on top of some of our observations would be considered pretty mystical as well if they turn out to be wrong!)

A great primer on what you can learn just by looking at the sky and reasoning (without aid of any tools more complicated than a stick to serve as a gnomon for a sundial) is Thomas Kuhn's The Copernican Revolution, which is in part meant to be a vindication of the pre-Copernican approach to the sky, even though it was not a strictly correct one.