As I understand it, the issue is this. I make a Twitter account. (I dunno why I want to call them "feeds.") I mark it private, which means that only people I allow to follow me can see it. Then, someone else downloads this app which then shares my private Twitter feed to the app makers without my permission.
So let's say Alice makes a Twitter account, and marks it private. She allows Bob to follow her. Bob then downloads this app, which can then see her tweets as she's allowed Bob to see them. Alice is willing to let Bob read her tweets but doesn't want them shared with other people, and Bob presumably doesn't realize he's violating her privacy by downloading the app.
At least, that's my understanding of what people are upset about.