How do I convert layers with blending modes on top of a black background layer to having a transparent background without losing the blending mode?

If you don't like the darker outer edge, you can add a blank layer above your explosion layer then select it and your explosion layer and press ctrl-e to merge them. then reclip all the color layers to the merged explosion layer. open up the layer effects dialog and add an inner glow with white, set to screen. adjust to your liking.

Clip your color layers to the explosion layer, making sure they're set to color blend mode, or you can merge the three color layers into one layer, like the video and then set it to color blend mode. Open the layer effects dialog by double clicking on the explosion layer and use the blend if sliders to eliminate as much of the black background as you want.

Take everything you have there and put it into a folder (the create new group at the bottom of your layers panel. looks like a folder.)
You then need to create a mask on the folder layer of the actual explosion mask (the white thing you painted...). I'm not sure where you have that, but it can be as easy as a command+click on that layer with the explosion inside of the folder, then applying the mask to the folder it's self. But without being able to see your layer stack, it's hard to say exactly how to get it there. A screen shot of your layer stack would really help answer your question more fully.

Hey!

On the layer with your explosion brush, CTRL+click on the preview in the Layers panel. Click on your black background layer, then click on the Layer Mask icon at the bottom of the panel:

That should do it.