The bulk of the increase in the number of cancer survivors alive today is due to two factors. First, despite what you may sometimes hear from naysayers, there actually have been some significant improvements in ways to both detect cancer earlier (usually at more curable stages) and treat it once it’s diagnosed.
The second reason is that we’re living longer overall. People aren’t dying as often of what doctors often refer to as “competing risks” such as serious infections like pneumonia. That’s also a large part of the reason why more people are being diagnosed with cancer, because cancer risk tends to increase with age.