You'll hear non-scientists saying that, but it's not quite right. Consider that right now, even as you're reading this, you are moving at 99% of the speed of light relative to some observer in some far distant galaxy. Do you think that time is going slower for you? Or that if that distant observer were to change his speed so that you were moving at only 98% of the speed of light relative to him, time would speed up a little bit for you?
What relativity does say is that if you can arrange to measure the time between ticks of a clock that is moving relative to you (and this is harder than it sounds) you will find that that while your clock, at rest relative to you, ticks ten times the moving clock will tick somewhat less than ten times. However, this effect is completely symmetrical - someone at rest relative to the other clock will consider hiself to be at rest and find that your clock, moving relative to him, is the one that is running slow. Thus, time does not "get slower for you" as you go faster - instead, you find that clocks moving relative to you run slower than a clock at rest relative to you.Again, that's not quite right. As far as you are concerned, you are at rest and everyone else is moving relative to you... and it is not possible for some other clock to be moving at the speed of light relative to you, so the question never arises. It is true that a light signal is always moving at the speed of light relative to you (that's actually the basis of all of special relativity) but a light signal isn't a clock. There's an FAQ on this already:
https://www.physicsforums.com/threads/rest-frame-of-a-photon.511170/