The short answer is… yes! You do actually. Not because of biology, but because of physics. Time Dilation is the fact that the faster we move, the slower time is.
Imagine you’ve rented a space car for two weeks which can travel to our nearest star system, Alpha Centauri in about 18 hours. And by the time you come back to Earth after your 2 week vacation, you cop a fine for “returning the car three weeks past the due date”. You’ve also been fired from your job because your boss claims you haven’t shown up for work in about five weeks!
Somehow, because you travelled to Alpha Centauri so fast, time slowed down for you! So what you thought was a 2 week vacation, was actually a 5 week vacation!
Physics is a bummer... but at least you’re 3 weeks younger according to physics! How can the world be like this? The short answer is... it’s Einstein’s fault.
Einstein famously claimed that the speed of light is constant in all perspectives. What does this mean? Well…
Imagine your friends are in a car which is moving at 100 km/hr and you’re watching them drive past on the sidewalk because there was no room for you. Then, you see your friend throw a banana out the window forward at 30 km/hr. From your POV on the sidewalk, the banana will look to be moving at 130 km/hr (the speed of the car + the speed of the ball).
Now, instead of a ball, your friend in the moving car throws a beam of light instead which moves at the speed C km/hr. Normally, you’d expect to see the light move at 100 + C km/hr (the speed of the car + the speed of light) however, you don’t! You must see the light move at the speed of C km/hr since, according to Einstein, it must be Ckm/hr in all POVs.
This has crazy implications. Because speed is a factor of time and distance, and the speed of light is always constant in all POV’s, then a person’s sense of time and distance must change between you and your friend’s POV’s.
Voilà! This change in time is called time dilation - or the slowing down of time.
Now go do some laps around your local park. The faster you go, the slower you age!