My goal is to get the fantasy baseball data from my league sent to my desktop. When filling out the New API Key form, even when I choose "Applicaiton Type" = "Mobile/Desktop", it still requires me to choose a Homepage URL and a Callback Domain. Can someone please explain what these do?
"Mobile / desktop" is the type of application you are developing, not your own use.
You need to have a URL and a callback domain to get setup. The Homepage URL verifies your website and the callback domain is where Yahoo redirects your users after they accept to let your application use their data.