How did Greenwich become the center of the world time?

When the Greenwich observatory was set up in 1675, its primary task was not to keep track of time but to solve the problem of fixing longitude at sea. The first thing its expert custodians did was to verify a long-held assumption that the Earth’s rate of rotation was constant. To check this assumption, the experts brought two newly invented pendulum clocks. This is how time-keeping at Greenwich began. Using these clocks as a benchmark to prepare time-wise star maps night after night to detect any shift in the star positions, the experts established (somewhat erroneously) that the Earth’s rotation is constant. They prepared navigational charts with Greenwich as a reference point for all longitudes.
Greenwich Observatory

Many great minds were bent on establishing the longitude’s relation with time. While the latitude was determined by the altitude of the sun relatively easily, it took centuries to establish the longitude. The position of the moon and planets, particularly Mars, was used to divide the earth into hours. The latitude naturally had equator as the starting position, but as the longitudes completed 360 degrees, there was no natural starting point. Countries used their capitals as reference points.

By 1770, all navigators were using them. With the development of the United Kingdom, Greenwich was unofficially used as the standard for the measurement of time. Sailors and travelers kept one chronometer at Greenwich to have the reference for time. In 1884, at the request of the US then-president, the international Meridian conference was held in Washington. The objective of the conference was to determine a prime meridian for having a set location for zero or standard time. Ironically, the choice was Greenwich, situated in the United Kingdom. Partly owing to the observatory and party for its history of being the reference point for the measurement of the time, it was the obvious choice. Declared in 1884, Greenwich became the world’s reference point for the measurement of time. The countries’ times were considered as GMT plus 4, or GMT -4, for instance. 

Years later, in the 1940s, it was realized that the Earth has been slowing down and speeding up irregularly. Hence, the night sky relative to the Earth also does not remain constant. In 1986, Coordinated Universal Time/UTC, based on atomic measurements, became the world’s preferred time. It is coordinated from Paris and is based upon an average of clock readings from 25 countries around the world.