Last week's lesson was time dilation, and you hopefully understand the basics of how it works. Cliff notes: According to Einstein's theory of relativity, the faster you travel, the slower time passes. So if you get in a spaceship and fly away, so from the perspective of a person on Earth, your spaceship's clocks would be slowing down.
But because everything is relative, you can flip the whole thing. From the perspective of the spaceship, the Earth is moving and its clocks are running slower. Both of us, you on the spaceship and me on the Earth, would see time passing more slowly for the other person. Each of us would think the other person's clock is slower.
But this raises an interesting question: what happens if you turn around and come back to Earth, and we compare our clocks? Whose clock is really going slower? The clocks on the spaceship. Why? I'll let explain:
The key is that spaceship has to turn around to get back to Earth. As it does, it has to change speed and direction, during which time the clocks on the Earth start to speed up.
So even though you'd see clocks on the Earth going more slowly for most of the trip, for the brief moments when you're turning around they'd be going much faster than yours, enough to make up for the difference in the twins "paradox." By the time you make it back home, your clocks would have fallen behind and you'd actually have aged more slowly compared to people on Earth.
Physics is weird, man.