One of my faults is a dislike of repeating myself. That doesn't work well with the blogging and education format. So I will repeat and clarify my previous thoughts.
Most of what you can easily find on the web about relativity (both special and general) is bunk and half-truths. As Terry Pratchett said, it's "lies for children". I have guesses as to why this is so. Most people, and this includes scientists, don't actually know much about what they're talking about. Scientists don't know much about the real world any more. They are mathematicians working with computer models. And they're not taught to write well.
So, here is my explanation of relativity, garnered from years of layman research and thinking.
You cannot explain relativity without explaining gravity. You cannot explain gravity without explaining quantum fields.
A quantum field is how each of the forces that comprise the universe exists and operates. Each force has its own field. Particles are standing waves in those fields. The fields interact with each other constantly, according to specific laws. Changes in the electric field produce changes in the magnetic field, and vice versa.
Spacetime is a field. It governs motion. Gravity is a differential in that field. Everything tends to move from a higher level to a lower level, gaining energy as it does so. The base energy of the spacetime field is a large positive number. All other fields derive their energy from spacetime, reducing it accordingly. This reduction propagates spherically at the speed of light. (Since the fields have been around since the beginning of time, most of this propagation has already happened. We just see changes now.) Gravity is an effect, not a cause. And it pushes from empty space, instead of pulling towards concentrations of energy. (Although I will admit, that is a bit of "Six of one, a half dozen of another.") The spacetime field, as all other fields, is quantized. ('Quantized' is a fancy term for working with integers, not real numbers.) In fact, spacetime quantization may be the cause of all other quantizations. There is a minimum bit of energy, and a minimum bit of time and space. But these minimum distances and times are relative to a particle. Spacetime is not a static, integer grid, but rather a smooth flow of the real numbers.
Look at the curvature of spacetime. That is to say, the difference in energy levels across an area. If you take a reference from a flat plane parallel to the resting energy of spacetime, you can see the angle made at each point. The sine of this angle is the motion associated with this spot, where 1=c, the speed of light. The cosine of this angle is the time associated with this spot, with 1 being the slowest passage of time possible. Note that these numbers are what a particle at that spot would experience, based on the spacetime curvature. Please note the elegance of this. Sin^2 + Cos^2 = 1, as Einstein proved the relationships of Special Relativity are based on the Pythagorean theorem.
Be aware that an alternate explanation is that the angle could be derived from a simple fraction of the whole - how much energy remains at that spot. This has a subtle but important difference from local curvature. This distinction could be tested by conducting experiments with atomic clocks at the various Lagrange points, most particularly L4 or L5.
What I just described is special relativity. This is the laws of motion and experienced time for a single moment in a static universe. General relativity is what you get when you turn the universe back on, and have constant movement and change everywhere.
Please note that any particle gains energy as it moves down the spacetime slope (stealing it from spacetime), and loses energy as it moves upslope (returning it to spacetime). Please also note that particles maintain their own speed and perceived time by having their own forward-reverse energy differentials. (forward is lower energy, reverse is higher energy. There is not net gain or loss, but a lot of energy can be involved in creating the difference. That's how AC power works, by the way.) This explains red/blue shift and differences in perception between viewpoints.
Please also note that the numbers generated are perceived, not actual. There is a baseline of space and time, independent of every observer. If this were not so, then photons could not exist, move, and change. To a photon, all space is the same point, and all time is the same time. Where would they travel, and when would they change?
Please also note that spacetime has an exact meaning. Motion is space divided by time. Therefore, for motion to exist, there must be a field governing both space and time. And for that to work, along with the basis of quantum theory, there must be a minimum time and a shortest distance. There must also be a largest possible energy, along with a lowest possible (non-zero) energy.
There is no disconnect between relativity and quantum theory. There are no paradoxes. Infinities exist only in the mathematical descriptions. Scientists, being human, often forget (or never learn) that the map is not the territory. The math is not the reality.
Quantum randomization is the effect of quantized waves interacting over a smooth spacetime. Integer based distances interacting while moving on a real number line always round up or down, but which way they round is based entirely upon things you cannot detect, since all you can see is the integers themselves. Everything is deterministic, yet simultaneously random through the wonderful effects of mathematical chaos. Small differences can have major effects. Or sometimes not.