Deleting backgrounds is a function almost always best left to proper image editing tools in all but the most trivial of cases. So my best advice is to not try to use SC to do this job. SC makes it easy to configure a proper image editing tool so that you can launch it with one button press, and SC will reload a modified image and preserve any objects, etc.
So bottom line is, use a good image editing tool to remove background stuff!
Having said that, the new function under the "SpecialFx1" menu, labeled "Delete contents of selection..." and you would trigger that after you have made a selection. Normally you make a selection by simply clicking and dragging out a rectangular region, but you can also engage SC's more complex selection methods, include magic wand, from the Edit -> Selection Mode menu.
Now, while I have advised using a proper image editing application to remove backgrounds, there are two operations commonly done on screenshots (not photos) which SC *IS* good at, and many people don't know about. These are:
1. Splicing out horizontal or vertical sections of a screenshot. This is super useful for doing things like removing a banner on a web page screenshot, etc. And I use this a lot. SC has a nice quick function for this under the SpecialFx2 menu, labeled "Splice out.."
2. Erasing some text or small item in a screenshot, which sits on a solid background. Again this is very useful for tweaking screenshots. You'll find this function labeled "Patch text/contents" under the SpecialFx2 menu.
Both of these functions are fast and easy and blunt force actions that can very very useful when working with screenshots.