The thing is, in evolutionary terms the genes that kill you before procreation are actively selected against; yet those that kill you just as reliably later in life are passed on.
Well not exactly.
The prevalence of senility in all the other apes (except humans) begs to differ.
I'm not saying that Richard Dawkins is wrong, I'm just saying that he's simplifying a little bit for the purpose of an explanation, but reality always more complex in the tiny details.
The thing is, we human have invented one peculiar concept: the grandmother.
In most other species of apes, individuals don't serve a purpose once they're past their reproductive age. On the countrary, they are using up valuable resources that might be put to better use by the young and the individual that still reproduce (in the same pack/tribe/etc.)
Thus in most other species of apes, senile degenerescence seems to be actually the norm.
Past a certain age (not far from the end of reproductive life) most apes turn senile rather quickly.
There's a small advantage if individuals don't live too long after they stop reproducing, because it leaves more food for the younger individuals of the pack/tribe (individuals who share the same genetic mix - being the same extended family - and thus this is the special form of 'sacrifice' which might actually get selected for. Unlike 'lemmings suicide' urban legend). A gene leading to such situation will be selected for, because it leads to an increased number of individual carrying a copy of the gene, by optimising which individual keep reproducing.
Compare the situation with humans:
disease like Alzheimer's, Parkinson's, vascular dementia, Huntington's and other neurodegenerative and senile diseases are *diseases*. I.e.: special conditions that only affect a small proportion of the population.
Most individual go through their later years *without suffering* from any of the above (in stark contrast of the remaining apes).
Why so? Grand mothers (and grand parents in general).
In human specie individual who are past their reproductive age will help raising the youngest generation (their grand children and grand nephews).
They take care of the youngs and, once language has been developped, they can also pass their knowledge by telling stories giving explanations...
Even if an individual isn't reproducing anymore, and even if an individual isn't in their prime anymore, these individuals are *still* very valuable for the pack/tribe.
Thus there's a very light incentive to select for individual who can stay functionnal in their late years. Even if they don't directly pass copies of their own genes anymore, they do help indirectly the survival of the rest of the pack/tribe and thus helps indirectly that the extended family grows (which shares genes with them).
(it's similar to the type of indirect help that you see in a beehive/anthill. Most individual are infertile worker. But because they are all very closely related, by helping they increase the survival chance of other individuals carrying the same genes even without reproducing.)
So if you have children at 40 (disregarding the complications and risks) it's likely that they won't inherit genes that are likely to kill them in their 30s. Thus the population in western "1st world" countries is aging, having children later and this may also be a contributing factor to the phenomenon.
Also the *reason* while parents decide to have children later in life also plays a role.
Most of such parent usually decide to reproduce later in life because of *career* reasons: They want to be in a better paying position to be better able to afford the children.
This has the direct effect on the availability of healthcare and eraly diagnostics.
But has again a very slight effect on the family structure.
Chance are high that both parent will try to get back to their highly paid position after the birth, and thus grand parents might also again play a very slight role.
Thus families where the +65 y.o. are in better shape are a tiny bit better than families where the children are only raised by hired nannies.
Again the effect is very small and doesn't explain everything.
(lower pollution, lower man-made military-origin radioactivity, more elders trying to keep in shape with exercice and balanced diets also play roles on the overall effect)
But the fact that we as a specie have relied more on grand parents might explain a bit.