This is a simple one and many users have experienced the same. It is simply the way Wamp is configured out of the box. The JA Twitter module requires php_cURL to be enabled for it to work. While most webhosts set this as default Wamp does not. You have 2 simple options.
1: Enable php_cURL in Wamp. This is very simple. Click on your Wamp icon and go to the menu PHP>PHP Extensions anc tick the php_curl item. Then restart the wamp services and all will be fine.
2: Your second option is to simply logon to the back end of Joomla and unpublish the JA Twitter module.
Either will get you up and running in just a few seconds. Offline I prefer to disable the module as it can slow down the page loading while it searches for Twitter feeds.