The mad libs stuff is cool and I'm sure contributes to the increased conversions, but even without mad libs, I think when people see forms like the one on the left, it triggers "damn, another form to fill out" because they can easily count the different fields they will have to fill out. Then when they submit the form, in their mind (hypothetically), it would go into a stack of 1000 other forms and someone will eventually, maybe get to it.
The form on the right looks more like a message, in their mind (hypothetically) someone on the other end gets a text whenever it is submitted. They are also typing it as they read, so they're actively engaged as opposed to the traditional format, where most people know what is supposed to go into the fields without even looking at it.