I guess you're referring to time dilation - clocks tick slower when they're moving really fast.
I personally like Lorenz's thinking around this, which is (roughly speaking): There is some kind of aether in the universe and light is a wave in that aether, same as sound is a wave in air.
Every physical thing has atoms or electrons orbiting, and these are going at the speed of light - so that means when the entire object is moving faster, those orbits become squished because they can't break the light speed barrier.
This gives rise to two effects: Things seems to become physically shorter in the direction they are traveling (orbits are squished) and physical processes slow down (orbits are forced to move more slowly). The part about physical processes is important because clocks become skewed to a clock on earth and when you land, you will observe that your clocks do not agree.
I will note that this is considered a "fringe" perspective, but the more commonly accepted Einstein perspective is that "time just does that, get over it" is not very fulfilling...
