Yes, I have been meaning to document this.
The retweet option lets you retweet any of your friends tweets that contain the key words. Currently it only supports one set of keywords. I will look into supporting multiple sets of keywords.
Tweet search will execute the search string as a Twitter query, so will search all of twitter, similar to performing a search from the Twitter website. Your bot will process all of the tweets, and may answer any of the posts if it knows an answer, and is sure that the answer is correct.
When the bot decides to reply to a tweet is a little bit complex. The bots can reply in two ways, either from a script match, or from a known response match. For a script match (such as "What is 2 + 2?") the bot will only answer if the tweet was a question (to avoid spam). For a response match, the bot will only answer if its known response is > 90% correct. You can see the correctness of a response from the Chat Logs page, and can increase the correctness of a response by clicking the check mark button. Basically the more times the bot sees the same question answered by the same response will increase the correctness of the response.
I am working on simplifying the tweet search options.