The deal between Twitter and the two major search engines hasn't been disclosed but it's apparent that Twitter's public feed is of great value to both. It could be a real way for Twitter to earn some money and for Bing and Google's search results to be more realtime relevant.
The process is a bit complicated but what the search engines will do with the Twitter feed is index the tweets as they are popping up and scan them to check for patterns and spikes in keywords. This will help them get a better grasp on what people on the web are paying attention to and talking about. If a certain link is particularly popular, Google and Bing can take note and move the link or topic higher up on the results page.
The most daunting task for the search engines is that tweets will show up in the search results, but when there are millions of tweets coming in at once how do you rank them in the results? Twitter is apparently attempting to solve this issue, as are a few others to come up with a "pagerank" style of operating. But until that gets figured out by everyone involved I wouldn't expect a crisp and clear operation.