Special Relativity was discovered by Einstein and published in 1905, his miracle year. That year, he published 4 papers, each one a breakthrough. He got a Nobel Prize for one but it wasn't on special relativity. Still, Special Relativity is what started his fame. It's different from General Relativity, which he came up with 10 years later. General Relativity accounts for gravity and acceleration where Special Relativity leaves them out.
Special Relativity sprang from the strange fact that the speed of light doesn't change no matter how fast you're going. That took a while to figure out since light goes a billion feet per second. But, in 1887, some guys figured out a way to measure slight changes. They used interference patterns. (Michelson–Morley) And guess what happened? Nothing. The speed never changed.
It didn't make sense. It's as if we weren't moving. But we're flying around the sun at 70,000 MPH, that's 100 times the speed of sound. Then again the Earth is spinning and we don't feel a breeze. So they thought maybe we're dragging the light around with us just enough. They called it the Luminiferous Aether theory. (Gotta love it.) But it had plenty of its own issues.
Einstein thought let's just say the speed of light is constant and it's space and time that are squishy. This had all kinds of strange effects. For example, if you see 2 things happen at the same time, a traveler might see them happen at different times. As crazy as that sounds, Einstein just went with it. To demonstrate how that works, Einstein was famous for using thought experiments. A popular one is the light clock. Let's say you could get light to bounce between 2 mirrors a foot apart. And let's say you could see that light and count how many times it bounced. In a second it would bounce a billion times. Now if that clock flew past you at half the speed of light, it would move half a foot forward between each bounce. So the light has farther to travel. And since the speed of light is constant, it would take slightly longer to reach the other side. So it would bounce less often. In this diagram, I put a circle between each foot the light travels. For every 5 feet the the light travels, it only bounces 4 times.
So let's say your good buddy Ashley travels along with the light clock. Would she also see it bounce slower? No. To the traveler, the light still bounces between 2 mirrors a foot apart. So if it bounced any slower, then the speed of light is not constant. Instead, she sees it bounce a billion times a second just as if she and the light clock weren't moving at all. The only way that can work is if time moves slower for the traveler. That's called time dialation.
I have some news for you. All of that was thought of before Einstein. (FitzGerald 1889, Lorentz 1892) So what did Einstein add to this? Those predicessors depended on the idea of something not moving. But what is stationary? Are we stationary? Or the Sun or the galaxy? Einstein added the classic idea of relativity. Relativity, as a principle, has been around a long time. (Galileo 1632) A simple example is juggling on a jet. The laws of physics stay the same whether you're flying at 700 MPH or sitting on the tarmac with the engines running. In fact, without looking out the window, it's hard to tell what the plane is doing. Maybe a scale or an altimeter would help but not if you fly really low.
If relativity applies to speeds approching the speed of light, then your friend Ashley can juggle just fine. And short of looking out the window, there's no test she can do to determine her speed. For example, if she shined a flashlight in the direction she's traveling, us Earth bound observers would see the light travel out the front of the ship at the speed of light. Since the traveler is traveling so fast, you might think she sees it leave at half the speed. But no. Because of time dialation, she sees it travel out at the speed of light too. So that experiment won't work.
Now here's where the relativity part comes in. There's no reason why the traveler can't decide that she's the one who's stationary and it's Earth that's flying past at half the speed of light. And who are we to argue. There's no experiment we can do to proove her wrong. For example, if she looks at the light clock sitting next to you here on Earth, she would see it running slowly. Just to be clear here, you see her clock running slowly and she sees your clock running slowly. How can that be?