When did they start to "feel" cannot be answered (and does not relate to history in the strict sense. This certainly happened in pre-historic time. You cannot plan anything without some feeling of time. Perhaps some animals also "feel time", how can we know?)
But when people started to keep track, one can probably explain.
Not later than with the start of
agriculture. When doing agriculture, one needs to know the time of the year, when to plant. For this it is necessary to look at the sky, and observe various phenomena, and to count days. One has to know how many days remains to the next summer, and whether you have enough food till the next harvest. So one of the most fundamental thing for a human is how many days are there is a year.
So great efforts were spent to determine this with high accuracy.
Perhaps it started even earlier: those people who lived by hunting and gathering
also needed some notion of the season of the year. To keep track of the seasonable migration of animals, etc.
Some time after the invention of agriculture, writing was invented, and people started to record longer periods of time, count years etc.
By the way, all evidence shows that writing was invented to keep track of the
agricultural products and their distribution.
For shorter periods than a day, the Sun's approximate position in the sky was enough for
crude estimation of time in most civilizations. With the development of more sophisticated societies, more precise measurement was needed and water clocks