Why do screen icons start to jiggle every time I lightly anywhere on a screen? How do I turn this off?

When you hold your finger on the screen it switches to Screen Editing mode, where you can move the icons around and delete ones you no longer want to use. This support link describes how do edit your screen→Move apps and create folders on your iPhone, iPad, or iPod touch - Apple Support

Just touching the screen should not make that happen. You have to hold it for at least 2 seconds for it to go to screen edit mode. And if you touch on an icon (as opposed to the screen background) a menu will appear first.

You can adjust the sensitivity in Settings/Accessibility/Touch Accommodations.