I'm unfamiliar with the math of general relativity, but I have a pretty fair understanding of it. It starts with the equivalence principle, which states that the force you feel if you are accelerating is the same as the force of gravity. For example, if you were in a coma and woke up in a spaceship, you wouldn't have any way of knowing if the 1G force that was pulling you to the floor was the force of gravity, or just the force of acceleration you feel because the spaceship is firing its rockets.
One example of this equivalence you can experience in everyday life. If you hold a helium balloon you are aware that it automatically moves the opposite direction of gravity. If you hold the same helium balloon in a car, when the car accelerates you feel yourself pushed into the back of the seat, but you'll see that the balloon moves toward the front of the car.
So if the two forces are equivalent, what does that mean? Imagine you are in an accelerating rocket ship. If you have a flashlight, light emitting from it will appear to bend toward the floor, because the spaceship moves upward from its original reference frame toward the beam of light. If the equivalence principle is right, this answers the age old question of whether light bends under the force of gravity.
Now, how do we go from this to warped space and time?
Time is the easiest one for me to understand. Say I aim my flashlight toward the ceiling of my accelerating spaceship. If somebody above me looks at the light, it will be red shifted. This is because he is accelerating away from the place where the light was when it was first emitted, because the entire spaceship is accelerating. Because of this, each wave of light is going to have a larger gap between it by the time it reaches him than when it was emitted from me.
Now say he is looking at a clock instead of a flashlight. There are only so many light waves emitted from my clock each second. But if he is seeing red shifted light, that means the time between the waves is larger. The only way he can see what is going on with my clock is by seeing the light emitting from that clock. So if there's only a certain number of waves for each tick of my clock, and he sees more time between those waves, then he sees more time between ticks of the clock as well. As a result, my clock is ticking slower than his.
Similarly, if I look at his clock, I'm accelerating toward the light from his clock, so the light from his clock is blue shifted, the time between waves is reduced, the time between ticks is reduced, and his clock ticks faster than mine.
By the equivalence principle, if this happens in an accelerating spaceship, it happens in a gravitational field. So gravity warps time.
Gravity also warps space. This is harder to explain, because you have to consider the fact that gravitational pull is different at different distances from the earth. However, like I already said, by the equivalence principle, a beam of light has to bend in a gravitational field. (But just like in the example of the accelerating spaceship, the light isn't really "bending", but rather, our reference frame is accelerating to make it look like the light is bending.)
Since the pull of gravity is different at different distances from the earth, the light will "curve" by different amounts. Now normally, if you draw one line, and then you draw two lines perpendicular to that line, you have two parallel lines. That's what happens in flat space. But what if I draw a line from the center of Earth into outer space? And then what if i fire two laser beams perpendicular to that line? They will appear to curve toward the earth, due to the equivalence principle. But since the gravitational acceleration is stronger closer to the earth, the beam of light will bend more if it closer to the earth. As a result, even though both lines are perpendicular to the first one, they are NOT parallel lines! They are moving away from one another. This is not possible on a flat piece of paper. It is only possible in curved space. So space is warped.
Here is another way of looking at it. Like I said, the equivalence principle states that the force of gravity is actually an imaginary force, the same force we feel in an accelerating car. Therefore, while you sit in your chair, you are not actually sitting still. You are accelerating away from the center of the Earth at 9.8 meters per second per second. And objects that are in free fall do not feel the force of gravity, they are weightless. So objects that are in free fall are not accelerating toward the surface of the earth. The surface of the Earth is actually accelerating toward objects that are in free fall! And yet it is quite clear that the Earth is not expanding. If the surface of the Earth is not expanding, and yet it is accelerating away from the center of the earth, the only possible explanation is that time and space are warped in such a way that everything remains "at rest."
All of this sounds really weird but it has been experimentally verified. The universe we live in really is this strange.
