I started following the Pawtucket Red Sox that season with it in mind that I wanted to know who these guys were BEFORE they were called up. That is how I started following minor league baseball: to give me an edge on the average fan.
Being in New Hampshire, there are so many teams around; the Paw Sox, Sea Dogs, Spinners and Fisher Cats all within two hours. This made it very simple to see any minor league games whenever I wanted to, which became another factor: with so many minor league teams close by, it was easier to see the prospects in person.
So for me, my interest in the minors was an outgrowth of my interest in the majors, curiosity, and geography.
The Boston Red Sox and How I became interested in the minor leagues
Marisa Ingemi describes how she became interested in minor league prospects.