Transparency on vector Sparkle

I originally replied to this conversation without reading the first post that was made in the InDesign forum. Now that I understand that you are talking about an .ai file from a stock photo company, this is much easier to handle. The problem with the image you are seeing is that in order to show the effect of the sparkle, it needs some sort of background. Whoever set up the store where you purchased it decided the sparkle would be most identifiable on the default background in the Illustrator app.

When you are placing the image in InDesign, you need an .svg or .eps file. I have always used .eps, but the admin who closed this original thread in InDesign emphatically stated not to use .eps without any explanation. 

inside Illustrator, you can select File > Export > Export As... and choose .svg to save the file in a generic vector graphics file. Then import the .svg file in InDesign using place artwork. It should retain its transparency then.

My original answer didn't have the context that your original post or at least an almost identical post in InDesign had. I'll leave it in case someone else finds it useful.

Previous Answer:

I'm afraid that what you are doing is copying the rendered image from your browser. Unfortunately, when copying a bitmap .png file, the background is almost always black. But there is hope!

If you want to capture JUST the sparkle from the webpage:

If it is an .svg file, you should be able to look through the HTML in the web page using Chrome's developer console.  Point at the sparkle image and right-click to select "Inspect this" and it will open the developer console and you will see the sparkle (or something in the vicinity) selected. Isolate the beginning of the <svg> tag and select the entire <svg></svg> tag. Copy and paste into Illustrator. If you get a bunch of text in Illustrator, you grabbed the wrong item or didn't grab enough of the <svg> tag. 

If the sparkle is a .png file simply save the sparkle.png file to your computer. Then open it in a bitmap editor like Photoshop and it should retain its transparency. But you will lose the ability to edit the vectors.

All of this assumes that you have access to the HTML directly. Some pages are obfuscated or script-generated and there are lots of ways of keeping you from using the artwork on a webpage. If it's your original content, you should have the original .ai file and none of this matters.

Hope this helps.

Please see this video I made about it: https://youtu.be/Tiqm5hdQA5A