This is the most mind blowing fact in all of science, in my opinion. Put simply, the speed of anything is the distance it travels in a given amount of time. So, if you alter the distance travelled or the time spent travelling that distance, you get a change in speed. Well, it turns out that when we’re talking about something as fast as light, the distances and time actually do change so that the speed of light stays constant!
So, the driver of a car travelling 100 mph sees light leaving his headlights at 186,000 miles per second. You, standing next to the road, also see the car go by at 100 mph and the light leaving his car at 186,000 miles per second, but for you time is going slower, and/or a mile is longer than what that driver sees.
In terms of human perception, you still think the car is going 100 mph, because the the alteration of time and space for you and the driver of that car are extremely small at a relative speed of only 100 mph. There is a difference, but you can’t really measure it. However, if you put a very accurate clock on a jet and send it around the world at high speed, then compare it to an identical clock which was left on the ground, then those two clocks will show different times by some tiny fraction of a second. It’s always the exact same tiny fraction of a second predicted by the math used to determine exactly how much time should go slower and distances should shrink for that clock on the airplane. The clocks on satellites orbiting the Earth at tens of thousands of mph experience a significant enough time and space dilation that the difference becomes very noticeable and must be accounted for when syncing their clocks to operations on the ground.
If we could build a spaceship which travelled near the speed of light, the changing size of space and rate of time would be very noticeable.
So, yes, you and that 100 mph driver both see the light exiting his headlights and covering the same amount of distance in the same amount of time. However, if you could measure that distance and time very, very precisely, you would find that you disagree about things like how long a mile is or how long a second is.
Any relative motion produces this effect. When you walk across the room toward a stationary person, your perception of how fast time ticks by is different than that person’s. The difference at that speed is just so tiny that you can’t even come close to noticing it.