Peter Watkins said:
Indeed, there never has been an explanation as to why galaxies are separating at a rate that increases with distance.
Sure it's been explained. There are only two possible behaviors for a universe that is, on large scales, homogeneous and isotropic (the same everywhere and in every direction). One is uniform expansion. The other is uniform collapse.
Uniform expansion behaves as you describe: recession velocity is proportional to distance.
So, why was our universe homogeneous and isotropic? And why is it expanding instead of collapsing? Well, inflation answers the first, and partially answers the second. Basically, inflation is a period of fast accelerated expansion driven by some high-energy field. This blows up a tiny patch into everything we see. So it doesn't actually matter what the beginnings of inflation were, as long as they were such that inflation could begin. And once it did begin, the inflating patch became almost completely uniform in the blink of an eye.
This also partially explains why we're in a period of expansion, because inflation is itself an expansionary phenomenon. A period of expansion is therefore necessary. How long this period of expansion lasts, however, depends upon the contents of the universe. If the contents of our universe were different, it would be conceivable that it could be collapsing now.
So, though inflation explains the expansion, it does leave some unanswered questions:
1. What, precisely, is inflation? Though we are fairly confident that inflation occurred, there are still some big questions as to its precise nature. These are being tested, however, and it is hoped that future CMB experiments will have quite a lot to say on this front.
2. How did inflation get started? We have even less idea here. If we're very lucky, nailing down the precise properties of inflation will allow us to determine how it started. Since inflation itself hides how it began, it is unlikely we will ever obtain direct evidence of how it began. Hopefully indirect evidence will be forthcoming.
3. Why are the contents of the universe in the ratios they are? This depends upon a lot of as-yet-unknown properties of high-energy physics, as well as the mysterious dark energy (or modified gravity, whichever it is).
To sum up, yes, we know the answer to this question, but our answer raises still more. Such is the nature of science.