I’m going to try a different thought experiment than the others.
Imagine your clock is an LED, a light sensor, and a mirror. The “tick” of this clock is: the LED sends out a brief flash of light, which travels to the mirror, bounces off, and returns to the sensor.
If this clock is stationary relative to you, the time of that tick is the distance between the three components divided by the speed of light (c = d / t).
If it is moving relative to you, the light has to be at an angle, shining at where the mirror will be by the time the light gets there. This is a longer distance, and so it ticks more slowly from your perspective.
Every atom in your body, or in a clock, is held together by forces that behave like the light in the example clock.
Give two people clocks and have them move relative to each other, and both think the other is slow. This leads to the “twins paradox”, and while I can say the answer involves acceleration, I don’t fully intuit it and can’t help you with it.