This concept is called time dilation and it is a part of Albert Einstein's theory of relativity. Basically, it states that the passage of time appears to slow down for a faster moving object relative to a point of view at a lesser velocity. This is not due to any kind of mechanical issue or optical illusion, it is part of the fabric of space/time itself. But both individuals in both frames of reference will experience time as normal within their own frame of reference. The faster moving person will not feel like they are in slow motion.
Besides velocity, the other factor that influences the passage of time is gravity. An increase in gravity makes time slow down. Both of these factors affect astronauts in space but in opposite ways. The astronauts who spend months on the ISS are travelling at a greater speed while falling through space than people on Earth. This slows down their time. But they also experience much less gravity than people on Earth, thus speeding up their time. Ultimately, the velocity factor wins, and when comparing the atomic clocks of the astronauts who spend six months on the ISS versus the clocks of mission control on Earth, the astronauts have aged by 0.007 seconds less that their Earth-bound counterparts.