How do I do this Text animation in Sony Vegas (..or AE)?
It's called Kinetic Type generally (that link is for Templates). Depending on what you're looking to do you either need to do it manually or can use scripting to auto-fill a list of words. If you auto-fill with a list its generally more randomized, so if sequence is important. Like you want the Lyrics to appear in the order they're spoken you must do it manually.

You can use After Effects or Flash to create something like this.


Tools to help make Kinetic Typography (Video Stack Exchange for After Effects)
Using a Script to do it automatically with random words (Levitated.net for Flash)