How have people throughout history used shadows cast by the sun as a way of measuring time?

Before the invention of clocks (and even for some time after) people mostly relied on the shadows cast by the sun to keep track of time. Using shadows to tell solar time involves keeping track of the position of shadows in relation to a stationary object. Tools like a sundial, shadow stick, or even simply the shadow cast by a tall building can all be used to measure time. Bear in mind that these means of telling time by shadows tell us the solar time, which may not match up perfectly with our standardized systems of time. In solar time, the Sun reaches its highest point in the sky at midday, but this isn't always precisely at twelve noon. Another important thing about telling solar time is that it can only be done during the daytime! Of course, most people throughout history lived agricultural subsistence lifestyles and didn't have much concern for the hours of night. 

In Ancient Rome, sundials were a favored method of telling time. Some sundials were installed in public places so that any person nearby could approximate the time, which the Romans split into twelve hours. A few wealthy families even kept sundials in their homes, but most people either relied on public sundials or making an estimate from the shadows of nearby objects. Some people who worked in the fields or otherwise did not have ready access to a sundial may have used methods like this one to approximate the hours before sundown based upon the position of the Sun in the sky.

All of these methods rely on the apparent motion of the Sun through the sky. We now know that the Earth and Sun are both engaged in constant motion, with the Earth rotating on an axis and revolving around the Sun. This creates differential lengths of days during the revolution and the experience of individual days and nights as we turn on our axis. To us, it looks like the Sun is rising and setting above us, and the shadows cast during this journey may be used to approximate the time of day. In the morning and evening, shadows appear quite long due to the positions of the Sun and the stationary object. Approaching midday, shadows seem to grow shorter and may even adjust to the right or left. At midday, there may appear to be no shadow at all, and then throughout the afternoon, shadows lengthen again.

You could try to experiment with telling time on your own, even without a sundial. When you are outside during the day, take note of how long or short your shadow is. You can make a guess to the time and compare with the solar or standard time for the day!