The graph idea is the best way to understand it. Consider a universe that only has one dimension of space, a line. We need to use lower dimensional analogies, since no one has invented any four dimensional paper that I know of. Now, draw a whole series of horizontal lines, each above the other. These represent the 1D universe at different times. So, we have a two-dimensional spacetime (or 1+1 dimensional, if you prefer.). Now, draw a dot at the center representing some observer. This graph represents his unique frame of reference - it encodes the velocity of observers relative to him, the distance in between objects, the time in between two events, what events are simultaneous, etc. Since an observer is always at rest in his FoR, we draw an arrow from this observer going straight up. He is moving through only time, not space.
Now, consider an observer moving with a constant velocity relative to him. Since his position is changing at a constant rate through time, his path is a diagonal line. Another observer who is accelerating would have a curved line through spacetime. Now, we can convert to the FoR of that observer who was moving at a constant velocity. Now the observer who was at rest before is now in motion. And since this is the FoR of the constant velocity observer, he is of course at rest. This is a Galilean transformation.
Next, let's go back to the first FoR. Imagine that two rays of light are emitted from the origin. Let's scale the units so that light makes a 45 degree angle through each box made by the intersection of the time and space axes. Since light is a constant speed for all observers, it must always trace out this same 45 degree angle, even when we change reference frames. Let's do this.
If we change to the second observer, we're going to need to change his coordinates, so that the light rays trace out the 45 degree angles. If you actually take the time to draw this, you'll see it - you need rotate his spacetime, so that the boxes are now rhombuses. Now, light remains the same speed. But at a price - consider two events that are simultaneous in the original FoR. That is, they're on the same line. Transfer this over to the new, rotated spacetime. Now, we see, because the lines are rotated, that the events are not simultaneous - this is the relativity of simultaneity - observers moving relative to each other disagree on what events occur at the same time. We can see another effect, too. Two events separated by a unit of time in the original FoR separated by less in the rotated the spacetime. This is time dilation, in which observers moving relative to each measure different amounts of time in between events.
Since time and space can be rotated into each other in this manner, we have to recognize they are two of the same entity - spacetime. Next, we can see that even though observers disagree on length and time separately, they agree in the total distance through spacetime, the invariant interval.
