An API is a technical solution to access a service, like Bettin.gs, from the outside. This means you don't have to go to Bettin.gs to add your bets or read your data, but instead build your own platform to add your bets from or to show your stats in whatever way you find best.
An API has a set of rules you need to follow to make everything work and you need some bit of technical understanding to use it. To integrate it on your own website or app you need to write code for it or hire someone to do it. You can also use a tool like Zapier to automate things like backing up your bets, automatically share your bets to Twitter or Facebook, create a blog post or automatically tracking tips you receive via email.
You can read more about what you can do in other help articles or on the Bettin.gs blog.
API stands for Application Programming Interface.