"Intuitively then, here we are far away from the "centre" and yet if we look back towards the origin we see light that is still heading towards us. Therefore, in my ignorance, it seems that we travelled here faster than the light."
The reason this intuitive understanding is wrong is, annoyingly, tricky to explain, at least at an intuitive level. The best I can come up with is to repeat the analogy of blowing up a balloon. Now for this analogy to work you have to imagine that the Universe exists only on the surface of the balloon. Anything inside it is not a part of that surface, anything that is not on the surface is not in the "universe".
So, with that picture in mind, imagine what an observer on the balloon will see. As the balloon grows, and grows, and grows, then things on the surface will appear to be moving away from each other. On the other hand, he'll be unable to find the "centre of expansion", because the centre is somewhere in the middle of the balloon, and not on the surface, and so not in the Universe, so he cannot find it explicitly without leaving his Universe behind.
We can do even more with this analogy too. Firstly, any light that was created when this balloon Universe began will be trapped on the surface and so will expand with our observer as the Universe grows, so there's no problem of moving faster than light away from the source of the light. The source is everywhere, in his world view: all light moves about on the surface, and so all light created at the beginning of the Universe will stay on the surface and stay visible.
Secondly, one can perhaps see that if you put two dots near each other on the surface of the balloon, and then blow it up very first, that a) the two dots will appear to move apart, but that b) this won't be as fast as the speed the balloon increased in size. This difference can be measured, if you want some numbers then if a balloon initially 10cm across grows by a metre in Radius in a second, then two points separated by one degree would have been initially separated as measured on the surface by about 2mm but are now separated by about 2cm. So, while the speed of expansion of the balloon was 1metre per second, the speed at which the two dots moved apart as measure on the surface is not quite 2 cm per second = 0.02 metres per second, so that it's a significantly slower rate of separation. On the other hand, two dots on either side of the balloon, as measured again on the surface, will have separated by an apparent speed of about 3.1 metres per second, which is obviously much faster than the rate at which the balloon was growing!
I don't think that anything changes much as the speed of expansion of this balloon grows to high values, either, so that even supposing the balloon expansion could only happen at the speed of light then it ought to be clear that two points either side of the balloon will appear to be moving away from each other at a speed many times faster than the speed of light. This also allows us to explain the whole "meta-universe" idea, that there is a huge amount of the real Universe we cannot see. We can't see because in some sense it was on the "other side" of the expansion.
The whole point of the balloon analogy is to try to emphasise that all of the physics the guy living on the surface of the balloon does, must be done on the surface of the balloon. That, for him, is the Universe.
I hope very much that this analogy is helpful and accurate, although if you have any more questions then feel free to ask, or to read some of mibn's links.