Not to take anything away from the worthy Canadian mentioned elsewhere in this thread, but the knowledge that local time differed in different parts of the wold dates to about the third century BC. I wrote about this more in this answer.

To quote a bit 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.

Sir Sanford Fleming, a Canadian, figured it out.

The necessity was brought about by the need to make sure there were never two trains trying to use the same track at the same time. Before trains, exact times weren't needed for any aspect of commerce or daily living and people just used the day/night cycle to set clocks locally. Many decent sized towns had a central town clock that everyone could synchronize to.