I use XAMPP myself, but WAMP isn’t much different.
I’ll bet you have not enabled the curl module. WordPress can use other methods (streams and fsockopen) as a fallback, but these may be disabled by default as well in a stock install. Curl is preferred and easy to enable.
- Close WAMP
- Go your the \bin\php\version directory in WAMP
- Edit the php.ini, and uncomment the line that reads extension=php_curl.dll
- Do the same thing in \bin\Apache\version\bin\
- Save and restart WAMP.
Edit: WAMP may also have an easier way to do this through the graphical interface. Look for “PHP extensions” and “php_curl” and make sure it is enabled.