Okay, this is what you do:
Right click the moving picture from this site's Sprite Gallery and select "Copy Image URL"
When you're making a question/answer, you'll see tool bar. The one with bold and italics and all the good stuff. The sixth button should be a square. Click on it.
It'll ask you to put in the URL of an image. Earlier, you had copied a Pokemon's animation from the Sprite Gallery, so paste that in. Then press confirm or OK or whatever it says.
You have your picture!
For example if you're taking Beautifly's Animated Sprite, then do the steps and you should have this:
(Beautiful choir music) (Beautiful Choir music)