Take a sheet of paper and draw two axes, time and distance. Then draw a line for a static object. This line would be parallel to the time axis, because for this object time changes, but distance does not. Now draw a line for an object moving with a constant speed. It would be a straight line on an angle, as both time and distance change.
Finally look at your drawing as a whole. It is static. Any motion that happened is already drawn as lines (called "world lines"), so now you just have a static picture of these events. This sheet of paper irepresents spacetime. It is static ("not relative" as you called it). No matter how you turn it, the picture on it doesn't change.
Now let's measure spacetime "distance" (called "inerval") on this sheet. Imagine some object moved by $x$ in space and by $ct$ in time (were $c$ is the speed of light used as a constant to make sure we don't add meters to seconds). In the "normal" (Euclidian) geometry, the interval $s$ would simply follow the Pythagoras theorem:
There is a problem though. If this were true, we could draw a line on such an angle that time would go backwards. This is not an option, because, unlike in space, we can't turn around back in time. To reflect this difference betwen space and time, we put a minus sign between them and write the spacetime interval this way instead:
This fixes the problem, but also makes this spacetime (called Minkowski space) counterintuitive. For $x=ct$ the interval is... zero. What does it mean? It means that time stops at the speed of light. Space and time cancel each other out.
Can we exceed the speed of light? No, but not because of the mass increasing (it does increase, but this is more of a result than the cause). We cannot exceed the speed of light, because time cannot move slower than not moving at all. It is like traveling to the North. Once you've reached the North Pole, can you move any "farther" to the North? No, your distance to the North Pole is already zero and cannot be smallr than that.
Can we move with the sped of light? No, but light does. What special property does light have? It has no mass. We have mass, so we can stay and move with different speeds, but we can't reach the speed of light. Light has no mass (called "invariant mass", formerly known as "rest mas"), so it can't stay or move slower without disappearing. In vacum, light can move only with the speed of light.
Why are time and space relative? What is the underlying reason? It is the fact that the speed of light is finite. If the geometry of spacetime allowed any speed, no mater how high (like in the classical Galilean theory), then we couldn't relate space and time to each other like above. There would be no spacetime intervals, only spatial distances and temporal periods separately. It is the speed of light that makes time and space relative and links them together.
Of course this explanation is simplified, but I hope it helps :)