Assigning numbers to any element of interest in our daily lives has become imperative today. Any scientific study is deemed useless without a reasonable quantitative base. Measurement: A Very Short Introduction provides a great history here for fields ranging from physical sciences & engineering to behavioural sciences to social sciences & public policy.
Time is arguably the most important, often also the base for many philosophical arguments. The history is particularly fascinating because unlike physically obvious quantities like length, time is rather abstract. Historically, many of the units of time have their origins in the planetary astronomical phenomena
One of the bigger drivers of the need of accurate measurement of time was driven by the need for navigation. Measure the longitude requires working out the time difference between noon at the Greenwich meridian and noon at our own location, revealing the angle between two locations, and subsequently the distance. This requires an accurate measurement of time; a clock which can keep time over long periods involved in sea voyages. This problem was of such importance, that prizes were offered by the state for cracking the problem, including one from King Phillip II of Spain in 1567, King Philip III of Spain in 1598, and from the UK Parliament in 1714.
Trade, surveying, and construction provided the initial impetus for developing measurement mechanisms as the civilization evolved. More recently, industrial revolution and scientific advances were two other major drivers of improvement in measurement accuracy of time, and other quantities in general.