The problem is that React is trying to access the result of the API call while it hasn't been fetched yet. You should add null checks when accessing nested objects (this is a javascript issue rather than something React-specific).
Secondly, while the data is unavailable, your component will still try to render something. React renders your component the moment you inject it into the page, so consider showing a "loading" indicator while the API result has not been saved to state.
Here is a fork of your fiddle with appropriate null checks & "loading indicator":
http://jsfiddle.net/jxg/9WZA5/
render: function(){
var degrees = this.state.item ? this.state.item.condition.temp : 'loading...';
return(
<div className="ww-container">
<div className="ww-current-condition">
<div className="ww-current-temperture">{degrees}°</div>
</div>
</div>
);