How do people stop DDOS attacks anyway? Is it like a separate box or proxy laid down in the chain of connected stuff that auto-ignores requests from any IP sending way too many requests at a much faster speed than the server could or something?
A lot of little strategies, rather than one big obvious fix. DDoS typically involves flooding of some type of traffic. A simple DDoS might be a flood of TCP/IP open-a-new-connection packets, designed to confuse and overload OS kernel networking software. Other DDoS's are simply a massive amount of valid traffic, i.e. sending HTTP requests to compute-intensive script on the web server, over and over again, hundreds of thousands of requests per second.
Each DDoS is different. The traffic sources may come from different parts of the world, originate from different ISPs. They may originate from a criminal DDoS black market, where armies of "zombie" machines may be rented by the hour to perform DDoS attacks.
One thing is certain, though: there is very little economic reason to pay DDoS ransoms, as that simply serves as a clear economic signal that you are a mark, and can possibly be taken for even more money. Paying ransoms encourages further DDoS. Criminal parasites don't need your business to be profitable and sustainable.
Typically a business will take unspecified technical steps themselves, or hire a security firm or DDoS-proof hosting firm to do it for them.
Sometimes it is possible wait out a DDoS, but that's not realistic for most web businesses/services. It could take weeks or months, as the cost of zombies is probably below the several-thousand-bitcoin payout that other thieves have seen in the bitcoin press headlines.