A trader works on the floor of the New York Stock Exchange.

Twitter's analyst day was a moment of truth for CEO Dick Costolo, facing growing pressure about slowing user growth and lack of a clear vision. Well he delivered that vision, and a number of changes to the product, and investors liked what they heard. The stock soared 7.45 percent after the presentation.

Why? Costolo's vision for Twitter to connect people to their world and to the world was clear—and he explained a range of ways they'll make the company better, and how they should make money doing so.