It might be better to ignore my comment and just watch this, but if you want to watch me try to contrive a train analogy, keep reading. Call me out if I messed any of this up; I'm just a university student.
Imagine we're sitting here getting data from two trains on parallel tracks. We're using a GPS to get the distance these trains have traveled and they're counting the number of ties they pass over. But we have crappy GPS receivers that can only display latitude data. We happen to know some things about the trains beforehand thanks to a man named Trainstein: they can only move at one speed, they don't have brakes so they also can't stop, and (luckily due to our crappy GPS) they pretty much go due north on parallel tracks.
So we measure: 10 m, 20 m, 30 m, on and on. Meanwhile, the trains are counting ties: clack, clack, clack, on and on. Maybe there's 1 tie per meter or something. They radio us whenever they've counted off 36,000 ties. If they're traveling perfectly due north, we see that they've gone 10 km north when they hit 36,000 ties. Perfect agreement.
There's are problems though. The ties everywhere are perfectly fine, but train B's track veers a little east. They don't notice because the ties are fine: clack, clack, clack. But we get some funky results when they radio us. They're moving at the same speed over the tracks, so they both hit 36,000 ties at the same time. We see that train A is 10 km north of their last known position, but for some reason train B is only 9 km north. How did this happen if they experienced ties at the same rate? Train B traveled along the hypotenuse of a triangle and the northward component is shorter as a result. Some of their northward distance was sucked away to became eastward distance.
The analogy is kind of ridiculous, so here are the parallels (ha). Measuring northward distance is like time as measured by some person in whatever frame they happen to be in (using a watch or anything traveling with them); 1 km could be 1 hour. Counting ties is like time as measured in some inertial reference frame; 1 clack is 1 second (see what I did there with 3600 ties per kilometer?). Traveling due north is like having no velocity relative to the inertial frame. Traveling slightly east is like having some nonzero velocity relative to that inertial frame. The speed of the trains is the speed of light. Usually it's all northward ("timeward") but if you move, some of that is sucked away from the northward ("timeward") component and put into the eastward ("speedward") component.
Train A was perfectly due north, so the 10 northward kilometers it experienced agreed with the fact that it passed over 36,000 ties. Train B went slightly east, so even though our inertial frame time says 36,000 ties were passed, it only "experienced" 9 northward kilometers. Pretend the astronauts are just ticking clocks. A clock on the ground stayed in the inertial frame (the earth is spinning so it's not really inertial but... close enough. Pretend we stopped it's rotation) so its time age agrees with ours. The clock orbiting the earth is moving pretty fast relative to us (on the order of 7 km/s, and for months at a time), so the time it displays is not going to agree; it'll be less.
For those wondering why I picked ties to represent inertial time, the train moving slightly east sees more ties per change in latitude. You can invert that to make train B see a smaller change in latitude per tie, since a moving astronaut sees fewer seconds. Also, I rewrote this like twice to account for that, so if I accidentally left something in there that seems backwards, that's why.