If you want to output a transparent background image, there's a couple of things you'd need to do.
1/ Set the paper opacity to 0% as spyres suggested. You can either do this when starting your new painting ( Opacity -> 0% ) or when you've started your painting, going to the background layer, right clicking, choosing "Paper settings" and setting the opacity to 0%. Essentially this is making your canvas a transparent sheet. You'll be able to tell you have a transparent background as ArtRage will display a grey and white checkerboard where the canvas was previously.
2/ When you save your file for use elsewhere, be sure to use a format which supports transparency. I'd recommend PNG I'd say if you're going to use it on the web, as JPG doesn't support transparency. To export your painting to a standard file format, go to the File menu, choose "Export as Image", then select the format and save.
Hope that helps!