There are two types of time dilation: dilation caused by being near a large body, and dilation caused by traveling very fast relative to another observer. Relativistic time dilation plays a bigger role for astronauts aboard a space station similar to the ISS. Even though velocity and gravity produce opposite time dilation, in this scenario, time dilation due to velocity win out. From Wikipedia:
Gravitational time dilation is at play for ISS astronauts too, and it has the opposite effect of the relative velocity time dilation. To simplify, velocity and gravity each slow down time as they increase. Velocity has increased for the astronauts, slowing down their time, whereas gravity has decreased, speeding up time (the astronauts are experiencing less gravity than on Earth). Nevertheless, the ISS astronaut crew ultimately end up with "slower" time because the two opposing effects are not equally strong. The velocity time dilation (explained above) is making a bigger difference, and slowing down time. The (time-speeding up) effects of low-gravity would not cancel out these (time-slowing down) effects of velocity unless the ISS orbited much farther from Earth.
If you want a really non-technical explanation, you can check out this NASA webpage. I don't think that you have to understand university-level physics (contrary to what the author says), but perhaps I'm wrong. Here's an excerpt:
Think about two different people who are each moving at different speeds while each simultaneously measures the speed of a light beam. They will each measure the same speed of light, even though they themselves are moving relative to one another. The only way this can be possible is if one's standards of both length and time (what you use when you measure speed) depend on your relative speed. In fact, if you stand on the ground and watch somebody else fly past, you will say that their clocks are moving slower and that they have shrunk. The speed of time depends on your relative speed. This effect is called time dilation.
Note, though, that the effects are really tiny in this case:
The problem is that by relativistic standards we are still moving incredibly slow, only about 3E-5 times the speed of light (this number is usually referred to as beta). For small beta, the time dilation effect that causes clocks to move slower goes as beta squared over 2, or about 5E-10,which is a really, really small effect.
The author found a figure of roughly 0.007 seconds after 6 months in orbit.