So if you get a bunch of t.co traffic and you don't want to pay Twitter for statistics, the only way that I've seen that you can understand how you are getting traffic is to search for keywords relevant to your site and hope to find a tweet that includes the t.co link that you saw in your logs. You usually can't even search for the t.co link itself (unless it appears in the text of the tweet.)
Very annoying for people who want to study their server logs without paying extra money, but a great way for Twitter to monetize. (Even if users decide to include a bit.ly link to get free statistics within a tweet, it will still be hard to track down as explained above.)