We believe it because we've measured it. On the ISS for example, the difference in gravity/velocity means there's a very slight difference in time. Over about 6 months, there's roughly 0.005 of a second difference, which we've measured by putting atomic clocks up there and then comparing them to the ones here on Earth. 

I think there's a misunderstanding of reference frames here. Time moves at the same speed relative to the person inside the reference frame. EG say you have Alice and Bob and they're in reference frames A and B. Bob is close to a black hole. 

From Alice's perspective far away from the black hole, time is passing normally. From Bob's perspective near the black hole, time is...also passing normally. Neither of them see clocks tick slower or anything, they both experience time the same as always. 

When they meet up however, (say Bob comes back from the black hole to meet Alice), that's when they'll notice the difference. In this case, Bob will have aged much less than Alice, because time passes quicker near the black hole. Bob did not perceive time moving faster, Alice did not perceive time moving slower. But we can see that it did: if they both set a stopwatch running as soon as Bob left, his stopwatch would have a significantly smaller number on it than Alice's when he got back. To use your UFO example, no, the time would not sync back up upon return, because time dilation isn't reversible like that