The GIFs on Twitter are not actually true GIFs, unless I'm mistaken. They're just videos that get set automatically to loop. Every time someone uploads a GIF to Twitter, it gets converted to their format. In order to download it, you would need to download it from Twitter, which will come out in .mp4 format. You would then need to convert it to .GIF format. I believe there are websites that can do this for you.