What it means is that a list of blogs you are following (receiving notification when new posts are added) will be added to the bottom of your Blogger Dashboard page (if you're a Blogger member) and to your Google Reader (if you have a Google account). No need to save all those bookmarks in your favourites, no need to keep checking back for new posts. Easy peasy.
If you elect to be a 'public' follower of, say, "Snippets 'n' Scraps", your profile photo (in miniature) will appear there in the margin of my blog, with a link back to your blog, making it even easier for my readers to pop in to your place and check out what you're writing about. Right neighbourly.
From the blog-writer's point of view, it's about warm fuzzies, knowing someone is interested enough to read my meagre offerings. If you choose to follow privately, however, you'll still be told when I post, but I just won't know you're there watching .....hmmm.