For this guide, we will be using this Luxury Text Effect as an example.

Please note that this Graphics file comes available in both .EPS and .AI file format, but for this example we will be opening the .AI file in Adobe Illustrator.

If you have opened an .ai Graphics file in Adobe illustrator, you should be able to open the "Layers" panel, and tick off the visibility (the eye symbol) of the "BG" or similarly named layer:
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Voila! It's really as simple as that. You can now go ahead and export the graphic, and the background should no longer be present on export (meaning you will have transparency instead of a background)

Now, to export, go to File > Export > Export As...

The following windows and prompts will appear - adjust the settings as you desire:

And the final result (a graphics layer with transparency):