With the introduction of iOS 7 came lots of new features and changes. One change is the way in which you manually close an app.
The old way was to bring up that “quick app switching” bar at the bottom, put it in “jiggle” mode, and then carefully touch that little red circle to close the app. The new way still involves the “quick app switching” feature but now it looks different. Once you get to the quick app switching view, simply swipe up on the screenshot of the app you wish to close. I can walk you through it below.