The next trick is a theoretical possibility. I guess drawing what you need is easier because your image looks complex.
As said in comments there are numerous shapes which have gradients from black to some color. They are layered with blending mode which makes black work like transparency against black background. The other colors in the gradients are selected so that the blending creates the wanted combination. An elementary example:
This has only 3 ellipses which are filled with different (elliptical) radial gradients. Stacked with blending mode screen they look a light effect if the background is black. Against white the stacked result is quite the same as your dirt-like excerpt.
In the next image the gradients are edited so that black is replaced with fully transparent same color as the opaque color at the other end of the gradient:
Now the stack works as well against black and white (not claiming it looks light if the background is white). The blending mode screen cannot be changed because the color mix would be different.
Unfortunately I do not know how to perform this trick automatically. You must edit every item and have them as separate layers with right blending modes (see them in the original). If you can have a black background you probably have no gradient editing problem. But you must copy the needed items one by one and stack them in the right order with right blending modes.