Looking at the src of http://gifboom.com/x/771600be you can find the following:
<meta name="twitter:card" content="player">
<meta name="twitter:player" content="https://gifboom.com/p/771600be">
<meta name="twitter:player:width" content="480">
<meta name="twitter:player:height" content="270">
<meta name="twitter:player:stream" content="https://medias.gifboom.com/medias/0928a540746501330b2e723c91561e03.mp4">
<meta name="twitter:player:stream:content_type" content="video/mp4; codecs="avc1.42E01E1, mp4a.40.2"">
So it appears that Twitter does not in fact support Animated GIF rather sites are converting the animated GIFs to MP4 and then referencing them. I think the reference to Animated GIFs on the photo page could refer to the ability of Twitter.com to convert uploaded GIFs to MP4 though this does not apply to the Player Card