There's a little problem on explanation above... special relativity (warning: weird explanation incoming).
The speed of light is always the same in every referece system. All right, since atoms have mass they can't go to the speed of light, but think of normal everyday things, such as cars on highways.
Imagine first you're stood before the highway, you can see the cars go by, say the all go at the very same speed. You can see in one rail those going north (say) and in the other those going south. For you they're all moving at the same speed, but some in different ways, like 88 miles/hour.
Now imagine you're on one of the cars going north. For you the original observer is now going south at 88 miles/hour, and the cars in the same rail with the same speed are like frozen. Now, the cars going south are now traveling at 176 miles/hour.
I made a awesomely drawn diagram of this on paint.
It took me while to draw it, no complaining about it
Same as Case B could happen if you're on the cars going southwards.
The point is, that dependeing on your reference system (or what do you call rest), speeds can change, in this case, for example, cars originally going to 88 mph suddenly are going at 176 mph.
However, this is not totally true.
There is something called "the gamma factor", and it has to do with Lorentz transformations.... or in english, when you go near the speed of light, things could go weird.
Imagine you're on a train which is constantly accelarating, till reaching 99.9999% of the speed of light (remember things with mass can't go at the speed of light, it's impossible).
Common sense tells us that if you trow a ball on the train, and if that ball is going fast enough, the "total speed" of the ball (measured by someone outside the train) could be faster than lightspeed (say the train is going 1 mph below the speed of light and you trow the ball at 2 mph... common sense says the ball is now 1 mph above the speed of light).
I made another perfect paint illustration.
This is against relativity? How can something go faster than the speed of light?
The answer: It can't.
Kinda.
What's the simpliest way of speed? Say everything on the system is at constant speed, so we can say that speed is the distance traveled divided by the time it took. The more distance in the same time, the faster it goes. The more time it took for the same distance, the slower is goes.
There are no walls or something to stop the ball moving, but time, however... is relative.
From now on, forget about common sense and prepare for weirdness.
See, there are certain values on the universe the scientists agree to be constant, every single time. Such as pi, the physical constants (like the amount of heat produced by chemical reactions, the charge of the electron... and the speed of light). Notice time in no constant, it can be warped, like space too.
The speed of light must be constant in every reference system, no matter how fast you go, the speed is the same. This leads to counter-intuitive results, but that's another kettle of fish.
In this case, since the ball can't go faster than the speed of light, the ball's time must be slowed down. Why? Because for the ball not crossing the speed of light barrier, it needs to go slower.
You, who trown the ball, would see it traveling each time slower until it finally stops, in the moment it could cross the speed of light. For the ball, instead, everything is going ultra fast, and the whole history of the unverse are seconds for it. For the moment it stops for the outside world, the ball have already witnessed, in theory, an infinite amount of time.
Wait, what?
This is the weird way to prevent breaking the only true speed limit of the universe.
Now, you're also traveling almost at speed of light, so your time is also slowed down. In fact, a machine going almost at speed of light is perfectly suitable time machine, going to the future. Depending on how faster it went, thousands, or even million years could go by when for the inner time of the train, only days went by.
This is shown on movies like Interstellar or Contact.
It's not space becoming time per se, but it's the way in which time can be warped.
Also near blackholes, since the gravitational force is too huge, leading you to going near the speed of light, warps time like that... but maybe a video can be useful for this.
YouTube