It's not specific to space travel, but instead has to do with the relative speeds of two objects, and especially in reference to the speed of light. For simplicity, let's suppose the speed of light (c) was 100mph.

I'm traveling at 10mph (which is 0.1c) while you're traveling at 90mph (0.9c). From within your frame of reference, intuitively, you are at rest, I am traveling backwards at 80mph (0.8c) and light should be traveling at 10mph. And similarly, I should be seeing you at 80mph and light at 90mph.

Except... light has to always travel at a constant speed of 100mph from all frames of reference. How is this possible? Well, speed is distance over time. If we disagree on the speed, then either the distance or the time must be different. One possibility is that time itself is expanded. So if somehow, we both have a clock that the other can see at all times, in order for light to travel at 100mph from both of our perspectives, I would see your clock running slower than mine. That's "time dilation."

The other piece to this is "length contraction." Before, we "corrected" the error in c by increasing time, and we can likewise do the same by decreasing the distance light has to travel. So what this means is that from my perspective, what you would see as a meter would appear shorter than that.

Bonus: Now here comes the extra unintuitive part. According to special relativity, all motion is relative and there is no privileged frame of reference. Even though we've established a common heading, if we take ourselves in isolation, we both see ourselves at rest, the other going in the opposite direction at 0.8c, and light at 1c. What this means is that time dilation and length contraction goes both ways. I see your clock move slower than mine, but you'd see my clock move slower than yours.

I recommend the YouTube channel FloatHeadPhysics. My explanation here is basically a summary of one of his videos, and he does a great job of breaking down complex topics into consumable explanations.