Or, another way to look at it: we are living long enough now that we have a much better chance of contracting cancer during our lives.
It frequently takes a fair amount of time for the various micro-level problems in cells to develop into cancer, and then it takes some additional time for that cancer to get bad enough to kill you - which is why young people rarely die of it, and one reason why "death by cancer" will be rare when everyone dies of something else while they're still too young to have had much cancer risk.
1) There is more cancer today or the cancers are more lethal.
2) We are better at diagnosing cancer today.
3) We've eliminated (or dramatically reduced) a lot of the things that aren't cancer that can kill you.
My money is that it's mostly 2+3, and that those have brought about a bit of 1.