So, the important thing to understand here is that until relatively recently, there was no universally agreed-upon answer to the question "what time is it?"

Every person and place more or less determined their own answer to the question. (Anyone can use a sun dial to determine the local time, to varying degrees of precision.) If you lived in London, most people would have considered Big Ben to be the "official" time. I suspect that local church bells or railroad served a similar function in many small towns.

This wasn't particularly accurate, but the important thing to realize is that it didn't have to be. Nothing had to be synchronized across long distances, so it didn't matter if someone 10 miles away had a watch that was 3 or 4 minutes ahead of yours. (This changed during World War I, which artillery bombardments over many miles of front had to be synchronized carefully)

The invention of the telegraph changed this. Now, for the first time, it was possible to accurately synchronize clocks across long distances. In 1865, the US Naval Observatory began transmitting time across telegraph wires.

Later, in the 20th century, we developed radio-broadcast clock transmissions. Then, back in 1981, David Mills (my old boss) invented NTP, which allows computers to synchronize their times over the internet.

Opheltes cover it pretty well. I should add, though, that the advent of railroads was a big force for the creation of standard time zones and standard time; in the US and Canada they created them, in 1883. Not just for the convenience of passengers being able to schedule departures, etc. but the trains themselves had to have very clear schedules for the use of the rails, and railroad workers had to keep good watches. A railroad-use pocketwatch was regularly adjusted, or regulated, and the date it had last been regulated was scribed into the inside of the case. Railroad stations had prominent clocks. The Western movie scene of the man hearing a train whistle and checking his pocket watch was not far off the truth; in a small town the train station likely had the best-regulated clock.