After reading the article I took a look at their website. I even watched the 4 minute cartoon they made, it was funny, but ultimately didn't really tell me what they actually do. I guess I'm not their target audience.
Cloudflare is a CDN with extra bells and whistles.
A CDN is a way to off-load the bulk of the requests to your webserver by moving the content as close as possible to your end-users, thus reducing the number of hops required to get to the content, which in return increases end-user satisfaction with your product due to a decrease in page load time.
The theory is that if a user gets a snappy service they are more willing to spend their money, and so e-commerce sites and sites that tend to monetize their users in some way find benefits in using services like these.
I hope that explains it adequately. To label cloudflare a mere CDN is a dis-service to them but for explanation purposes it might as well be, I'm sure someone from CloudFlare is able to give a much better explanation of just why their offering is not just an ordinary CDN but goes much further than that.
