OK, the explanations are coming out now:
Cyberus Technology Blog - Meltdown
So, basically, an application running on an Intel CPU can completely break through the protection between user and kernel memory to read arbitrary kernel data. And it probably affects all of them made in the last twenty years.
No wonder they're rushing to get the fix out so fast.
And, with hindsight, the problem should have been obvious. Using the cache to leak information was the first thing I thought of when someone mentioned that Intel CPUs were reading data before checking whether it was allowed to. It just requires a little clever programming to exploit.