This is a tricky one, but I'll try to keep it simpler. I'm not certain that what I'm about to say is 100% consistent with the math of relativity, but it's reasonably close enought to at least understand why time dilation occurs.

With that out of the way:

First, the speed of light is not really about light. The fact that light travels at that speed is a consequence of the nature of photons. The speed of light is the ultimate speed limit of the universe - you could call it the speed of causality more accurately. In fact, in a sense everything travels at the speed of light - that might not seem accurate, but I'll explain more in a little bit.

Second, what is speed? It's how fast something is moving in a particular direction. All movement is directional - you don't just arbitrarily go fast. You travel along a path. Typically, we think of this path as three-dimensional. If you're flying in an airplane, for example, then relative to a stationary object, you're moving upwards at a certain speed, to the left or right at a certain speed, and forward at a certain speed. Add those together according to some relatively basic math, and you have your overall speed and direction of travel.

Third - what I just explained isn't actually correct, because it ignores something rather huge that all of us take for granted - time. You see, we don't travel a three-dimensional path. We travel a four-dimensional path. And that fourth dimension is time. For most objects traveling at the speeds most humans deal with, time is actually the largest part of our velocity. That's why time seems to pass at the same rate at the scales that you're used to dealing with - the differences between your time velocity and something traveling at 60 mph is so miniscule that it's impossible for a human to notice it.

Fourth - remember how I said everything travels at the speed of light? Well, that's why time dilation is a thing. Your total velocity through four dimensions must remain constant. So if you are traveling through space faster, then the only way for you to do that is to travel through time slower. So as you speed up, your experience of time slows down. However, because you have mass, you can never actually reach the speed of light. What happens instead is that as you get faster, the extra energy actually makes you more massive.

Incidentally, because photons travel at the speed of light, they don't experience the flow of time. From the point of view of the photon, its entire lifespan (from the moment it's emitted until it's absorbed by another particle) passes instantaneously and simultaneously. This happens because photons don't have mass, and so their velocity in the "time" dimension is 0.