The generated follow button HTML includes two pieces: a link to a Twitter.com profile page; JavaScript loaded onto the page to render the full follow button graphics, load a window when clicked, and log events for your analytics software.
It seems like the JavaScript portion of the follow button was not loaded onto your page.
View source on your page. Search for platform.twitter.com/widgets.js to determine if the HTML code snippet you entered is present on the webpage served to browsers.
Twitter’s widgets JavaScript requires Internet Explorer 9 or higher. If your SharePoint site has a X-UA-Compatible directive instructing Internet Explorer to emulate older versions of Internet Explorer the JavaScript may not load.