How to move sparkle star vector from Illustrator to After Effect

When preparing Illustrator artwork for video you should turn on View Transparency Grid and Pixel Preview in the View menu so you can actually see what is going on in the file. Each element you want to animate or separate in After Effects should be on a separate layer and appropriately named. It is possible to create that star with transparency using a gradient in Illustrator and just use the layer in After Effects. You just have to prepare the artwork properly. 

Cropped screenshots and screenshots that do not show the modified properties of the layers that are giving you problems are pretty useless for diagnosing problems. If you cannot figure it out, set up the View Options in Illustrator and solo the problem layer, show us the Appearance Panel and post a screenshot. Do the same with the layer in After Effects. Select the layer, press 'uu', and take a screenshot.

Here is the screenshot. The original vector file is comes with a checkerboard and grey background that i downloaded from freepik. 

Its hard to know what's going on without the AI file. Send that over if you can. But it looks like you've either rasterized the content in AI/ not interpreted its alpha channel correctly in AE. Gradients are not great when imported from AI to AE, sometimes blending modes don't translate well either.

Anyway, lets do a quick fix for you. Select the Star in the Timeline, open the Blending modes column, select Add Mode to remove the black and composite the star correctly.