1. It will help them secure major enterprise clients, probably the govt. or with ties to the govt.
2. Great selling point to institutional investors come IPO time.
3. Navigating foreign business opportunities.
Although I'm not a fan of this move by Dropbox, it is important to note that Stanford has hired Rice as a professor as well. No one is abandoning Stanford, and the hits that Dropbox is going to take are going to be minuscule in comparison to the upside. This is just the hard reality.
Rice does have a lot of experience that is relevant to Dropbox and students at Stanford and I'd just like to leave it at that because at the end of the day, connections + experiences that come from being Secretary of State trump pure meritocracy or idealism.