was this discovery made as the result of scientific discovery and astronomy by people like Pythagoras and Galileo or was it made by explorers who compared time etc... How was this discovery made? was it topic of argument? how did it become known as truth?
Time zones don't exist as natural entities. They were invented as a convenience for human beings, by human beings. (Specifically by railroad companies because one you start accelerating to certain speeds and sharing tracks, being off by a few minutes will kill you.) If you're asking, when did people understanding that "local noon" (e.g. the point at which the Sun is directly overhead) varies from place to place — this knowledge is ancient. It comes from an understanding of what it means to measure time (the position of the Sun) and what it means (either that the Earth is turning or the Sun is rotating, depending on your cosmology — either way, "noon" is always a moving position).
The reason, I think, that modern people get confused on this is that with timezones, and with clocks, we have "unhinged" our sense of time to make it somewhat more arbitrary, and not tied to celestial motions ("noon" no longer means "when the sun is directly overhead," but "when the clock strikes 12:00"). But for most of human history, time was about the Sun and its position in the sky. And if you think about that at all, you'll find that the notion that "noon" is thus different in many different places is an inescapable conclusion.
I wrote an answer to this a year or so ago that I've re-used when the question comes back up again: https://www.reddit.com/r/AskHistorians/comments/3uufv2/when_did_people_understood_the_concept_of_time/
To quote from that:
Hipparchus of Nicaea (c. ~190-120 BCE) seems to have been the first person to propose using a grid system to find the position of cities (and other places) on a globe, which implies an understanding of longitude. He built on earlier work by Eratosthenes of Cyrene (c. 276-194 BCE), who had mapped the known Earth, including finding its circumference. Hipparchus' method of finding the longitude of places was to use the differences in timing of lunar eclipses at different points on the globe to calculate the difference between local time of those points; the drawback is that there was no accurate-enough method of timekeeping to lead to useful calculations. Edit to clarify: The difference in local time between observed beginning and end of the eclipses would serve, essentially, as a way to understand the longitude between places.
Did the first navigators to circumnavigate the world know to account for the discrepancy in their timekeeping when they returned home?