Online these images appear with the background so that you can see that there is transparency - sorry Egg, nice theory but it's nothing to do with Illustrator or Photoshop.
I went to your here link and downloaded one of the images and I had exactly what it said a png with transparency and no background. If I copy the image directly from the page (browser save image as) I'll get the jpg version with a chequerboard, otherwise I press the correct download button (they make it hard to find it) and I get the png with transparency. The image with the checkerboard is not the image that is downloaded via the download mechanism.
My guess is that the OP copied/downloaded (save image as) from the page but didn't use the download button.
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