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.
So you’re looking at your iPhone and you want to close an app or two. Let’s double tap the home button to bring up the quick app switching feature.
You’ll see all the apps that are currently open displayed as a carousel of screenshots, plus a screenshot of your homescreen on the far left. You can touch them to quickly go to that app. But if you want to close it, simply swipe up on that particular screenshot.
You’ll see that app fly up and away while the other apps fall into place
Now for advanced users, you could try to close more than one app at a time by swiping up on 2 at a time using 2 fingers.
You could even close 3 apps at a time! Just think of this as “throwing them up and away” so whichever way you do that is your choice.
You’ll know all the apps are closed when all you see is the homescreen screenshot.
Just touching the screen or the home button again at this point will bring you back.
Good work!
SimpleiPhone 
Good info for a new user like myself