Your question is difficult to answer exactly, because it touches the question of consciousness, which is problematic in general. But the answer is basically this:
Theory of relativity describes the space-time as four-dimensional space that exists as a whole. You are described by a world-line, which contains all states you go through from birth to death. There is many slices, which describe "you" in different times. What you can see is given by what information reached your sensory inputs (your senses) and how they are processed by your brain. The brain creates a model of reality based on your sensory inputs, which is stored in space-like slice of your world-line in the space-time, which we call present. This is what "you" can see. So the reason basically is in how the brain processes the data it is getting.
Along you world-line, there is many versions of you at different ages and each version feels its own present and the versions more in the future also have memories of the past (because the brain stores the past sensory inputs). There is no single "present time" in the theory of relativity, only the space-time as a whole.
But you should't forget that the theory of relativity is still only a very well working model. Even within the theory of relativity, the question whether space-time exists as a whole, or whether there is "present state", which evolves in time, is rather philosophical than scientific, because there is no observable difference between the two.