Let's say I am the omnipotent God of a giant rubber balloon and two miniature men that live on it, which I'm going to call Adam and Steve. Adam and Steve have just finished their lunch date at a fancy bistro and are now just starting their journey home. At the moment, they live in opposite directions and about 3 meters away from the restaurant. The universal speed limit that I have set on this balloon is 4 meters per hour, and so that is how fast they will travel. Normally, then, you could expect their trips home to take about 45 minutes.
Now, I'm a vengeful and homophobic God, and I've decided to punish Adam and Steve for their "infraction" by making their journey more difficult. I eventually decide on inflating the balloon at a constant rate so that its volume octuples over the course of, say, 56 minutes. That means that the distance that Adam and Steve each have to travel doubles from 3 meters to 6 meters. But the ground underneath them is expanding as they travel! Using these facts, I can show that instead of taking 45 minutes, their trips home take 56 minutes (I had to use calculus for this part). Since Adam and Steve each travelled at a constant speed, I can calculate the distance that each traversed using multiplication: (56 minutes)*(4 meters per hour)=3.73 meters.
In other words, even though the restaurant and each man's apartment are currently 6 meters apart, each man only travelled 3.73 meters. It might appear that they violated my speed limit by travelling at an average speed of (6 meters)/(56 minutes) = 6.42 meters per hour; it should have taken them 90 minutes to get home, right? But in reality, they were both following my speed limit the whole time, and most of the apparent increase in speed is due to the inflation of this balloon world of mine.
Now, Adam and Steve are dismayed by this setback to their romance, but they nevertheless plan another date, this time at another restaurant right next to the first one. I decide to destroy their future using permanent and quicker inflation of the balloon. I get my magical balloon pump and use it to double the circumference of this balloon microcosm every 30 minutes. Now, no matter what, Adam and Steve can never reach each other: once each has travelled 30 minutes, there is more distance between them than there was at the beginning of their trip. Having contented myself with dashing their hopes, I destroy their Universe with a wave of my analogy wand.
The first part of this analogy concerns "normal" inflation, where the expansion of the Universe implies that we see parts of the Universe that are farther away right now than the universals speed limit would appear to allow. The second part concerns inflation which occurs at such a fast rate that certain parts of the Universe will never be visible to each other.
edited to add:
Yla, m52 is right (although I wouldn't quite the resolution that way; I think this issue merits a double post).
The universal speed limit of relativity is just that: a universal speed limit. The time rate of change of the distance between two objects moving in opposite directions is not a speed. However, if you are moving with one of the spaceships so that it appears at rest and then measure the time rate of change of the distance between yourself and the other spaceship, then you are measuring a speed. In Newtonian theory, these two quantities have the same value, but in relativity they do not.