Delete black edges on illustrator vector

The easiest way to remove this is in Photoshop rather than Illustrator - Photoshop is better for raster alteration, and Illustrator better for vector alteration.

When you open it in photoshop, use the 'Colour Range Selection' tool: 

>Select (top menu) >Colour Range >Use eyedropper tool on the darkness >Use the 'Fuzziness' slider to select the amount you want to remove or alter (the localised colour clusters tick box will alter your selection slightly). 

It looks like a gradient that has been expanded into multiple rings.

You could try using the Lasso selection tool to loop around sections of the gradient, avoiding selecting the sparkles, and then delete one section at a time.

Another thing you could try would be to double-click on the artwork with the black Selection tool to go into isolation mode. You may need to double-click more than once to drill down. You'll probably be able to drill down enough so that you can select just the wand and sparkles. Then you could copy them, get out of isolation mode, paste them and delete the original artwork; or you could paste them into a new document.