If you have configured the API at Twitter and completed all the fields in the widget within your WordPress admin - copying the details as described here http://www.planet-interactive.co.uk/simple-twitter-tweets/ and input your Twitter username.
Then it is probably because your hosting provider is over subscribed (has accessed too many Twitter Tweet streams within the 5 minute allocation) from their servers.
Please give it some time to see if this changes. Once access to the Twitter service is restored, your Tweets should be displayed on your website as expected.
Once this occurs just once, you will no longer get this message as your site will always default to the stored Tweets we backup to the database whenever the Twitter service is unavailable.
Please do check your settings in the widget, just to make sure, then you have to just wait ;(