Can time be interacted with? Astronauts come back to Earth younger than they would have been had they stayed on Earth for that same period of time. They are traveling so fast relative to the Earth that time slows down for them. Does that mean that the astronaut interacted with time? Does time interact with speed? And if so, does that mean time is made up of some kind of fundamental particles, like gravity has the graviton? What does time have?
 A: You don't have to be an astronaut traveling at near the speed of light to experience this effect. It can be measured quite precisely here on Earth and it has been measured both on elementary particles in accelerators and by flying atomic clocks around the world in planes. 
Humans are simply not used to it because the differences between the flow of time in different coordinate systems are too small to be felt by our biological perception of time. Our internal clocks are just not precise enough to experience the world as it really is (the same is true for most of quantum mechanics). Instead, we are experiencing a simplified version of it, in which having one crude clock that is independent of motion is "good enough". 
Thankfully we can visualize on our computers what the world would really look like if we had the right kind of senses to detect the actual details of a relativistic world. Look up MIT's "A Slower Speed of Light" game, if you want to get an idea of what's "behind the curtain"!
And while all of this may sound very strange, it's no different from what people had to deal with when we invented the microscope. Suddenly there were all these bacteria and single cell organism and all these details on common objects that we could never see before! Or think about the changes in our ideas when we suddenly had telescopes. It opened up a new world. Even a small telescope can make you see tall mountains on the Moon, and Jupiter has moons of its own, Saturn has a ring, Mars has all kinds of structures on it that nobody had ever seen before and there were new planets and millions of new stars, that the naked eye can't see. 
Relativity is no different, except that the "microscopes" that allow us to see its workings are better clocks and particle accelerators. 
A: As for "why", that is either simple or complex! The simple answer is that it is a consequence of the speed of light being constant for all observers. That means when travelling relative to each other they both measure time differently. This site provides a simple math explanation
OTOH, time is measured differently in different gravitational fields, and that is a lot more complex - General Relativity
