The answer is subtle, and I will have to go on for a little bit. At heart the question is asking about the nature of time. What is it that determines how we age, how much time we feel we have experienced?
Traditionally, (as formulated by Newton) time was considered to be some one thing which exists everywhere and is the same everywhere. Aging, or a clock ticking, or whatever you use to tell the passage of time, is in that scenario just a measure of that universal time. In special relativity on the other hand, this is not what time is. Time there is rather much more closely related to our notion of distance, than to some universal phenomenon. What our bodies, our clocks, etc. measure as their age or tick is this notion of distance within the larger four-dimensional space called spacetime. As you know, distance in space is not a universal phenomenon. If you travel from Montreal to Toronto, you cannot say that you travelled a certain distance. The distance you travelled depends on the route that you took getting from Montreal to Toronto. If you travelled via Miami, the distance you travelled is different than if you took the Trans-Canada through Kingston. Similarly, for time. Since the amount you age, or the number of seconds ticked off by the clock, are now a measure of this generalised notion of distance in spacetime, the number of seconds will depend in detail on the path that you follow through spacetime. Thus you who travelled far away and came back to your home at almost the speed of light will have travelled a different "distance" than your friend who stayed at home, and thus you will have aged differently since your aging is a direct measure of how "far" you went in this generalised distance.
To get the details right, you also need to know some of the properties of this generalised distance. For distances in space, the Pythagoras theorem tells you that the length of the diagonal of a right angle triangle is longer than either of the two sides. However for this generalised distance in spacetime, that is no longer true. The length of the diagonal can be shorter than one of the two sides. This finally results in the fact that the person taking the trip actually experiences less time (less aging, fewer clock ticks) than the person who stays at home. In other words, the roundabout trip is shorter in time than the direct trip, unlike the behaviour of distances in space, where a roundabout trip is longer. This strange behaviour of spacetime distances are clearly important for the details of the result, but not for understanding the general principle that time is a measure of some generalised notion of distance, and that therefore two people who go from one point in spacetime to another will have experienced different times depending on the route they took in travelling between the two.
Of course the ultimate question of why time behaves that way, why bodies aging, or clocks ticking measure this generalised distance, rather than measuring some all pervasive universal thing, cannot be answered except to say that that seems to be the way the world works. It could have worked the way Newton thought it did, but it doesn't.