Cheryl Klein from Scholastic just posted some really good advice on how to become a book editor.
The only thing I’d add to the internship advice is that if you need to work your way through school, as I did, colleges often have student editorial jobs. At BYU, where I went to school, I was the editorial assistant for the Humanities Publication Center. I also did an unpaid editorial internship, worked at the library, and worked at the University Press, as well as an off-campus job editing phone books and a job typesetting college textbooks at a prepress firm (that last was in Champaign, Illinois, while I was attending University of Illinois, so it goes to show those opportunities weren’t just available at the one school). If you can’t afford to take an unpaid internship, look for paid positions. They’re harder to find at the college level, but well worth it, and I was able to graduate with a little experience on my resume. This led directly to getting my first editorial job out of college.