That code for Dogecoin is based on Litecoin, and early versions of the coin incentivized block miners through a randomized reward system.
Born from a Reddit meme, Dogecoin had a ready audience of supporters ready to buy into the cryptocurrency, especially on the WallStreetBets subreddit.
Dogecoin is so cheap because there are so many of them, and because so many more are coming into existence, without limit, for the foreseeable future.
After opening an account and funding it, you can use those funds to trade Dogecoin or other cryptocurrencies.