This myth has been propagated by pseudotechies for as long as Apple has had the app switcher tray in iOS. When you switch to another app, iOS essentially puts the other programs on ice or parks them so they don't consume memory. Doing what this columnist shows only removes them from your recently used apps list and in the instance of a non-responsive app, force shuts it down when in active use. This was explained at launch of this feature yet this silly advice has been handed off as a tip over and over again.
@DerekBolander You're not totally correct. When you switch to another app, the original app goes into Background and then Suspended state. While in suspended state, the apps do not use processing power or drain the battery, but they are consuming memory. They will be purged and their memory reclaimed if needed by another app, but until that time the Suspended apps ARE still consuming memory. You can test this for yourself using an app like ActMonitor to check memory usage. For example, on my iPhone 5 I currently have 46.3 MB free memory. After doing what this tip suggests and clearing all the apps in the tray, I now have 650.4 MB free according to ActMonitor. It shouldn't make a difference since iOS should be purging memory itself whenever it needs to reclaim some for use by another app, but this tip does in fact clear the apps from using RAM. Whether that makes any performance difference or not is another story.
Have a look at the link I referenced in my other posting. Even when an app is in suspended state it is not slowing your phone down. iOS purges apps in memory as needed.