The Twin Paradox and Special Relativity - Say what?

I watched another video created by the Open University: 60-Second Adventures In Thought. I was quite intrigued but puzzled by Einstein's theories on special relativity and time presented  in the video. "The faster one travels through space, the slower they move through time?"  I do not know what to make of that statement. If a clock goes slowly it does not change the actual time, right? Maybe I'm not quite grasping Einstein's concept. So I went in search of information on the Time Paradox and Special Relativity's place in it. This is what I found. 

 In the natural science Physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into outer space in a lightning speed space shuttle and returns home to find that the twin who remained on Earth has aged more. This result appears somewhat confusing because each twin sees the other twin as moving, and so, according to an incorrect naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged more slowly than before. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey, and so there is no symmetry between the space-time paths of the two twins. Therefore the twin paradox is not a paradox in the sense of a logical contradiction.French physicist Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason...". Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se. Conclusions put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration.
The twin paradox has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment. It was also confirmed in particle accelerators by measuring time dilation of circulating particle beams.

I am still somewhat confused, but more informed than before.



https://youtu.be/oOL2d-5-pJ8