To copy your IP address, type copyip into the omnibox, then press tab and enter.
Copy your IP to the clipboard from the omnibox. Type "copyip", press tab, space and then enter. Similar extensions I've seen require you to click a button in the toolbar, which consumes screen space. Note that it may take a few seconds for the IP to be copied.
My IP is faked in the video. The extension will give you your real IP.
Privacy note: In order to obtain your IP, this extension calls a web service hosted on AppEngine using CORS over HTTPS. The data transmitted is the same data the browser sends to any web server when loading a web page.