What's the benefit of doing it this way, as opposed to selling some of their own shares from a previous round to the funding VC, or on the secondary market? Is it just a cleaner deal this way?
I don't think it's actually that common, and typically used with really hot companies.
A few reasons: it makes the new investor more competitive to the founders if there are others vying for the investment, but it also prevents selling too early.
If you have $5M in the bank, you'll be more likely to try to go for the home run rather than sell to facebook for $3B (which was rebuffed by the founders of snapchat).
Makes sense - it's sort of a psychological exploit to make the deal more enticing. Of course founders can always sell their shares on a secondary market, but it's much harder to turn down $5M cash.
One example of this is IVP leading a round in Snapchat, and the two co-founders splitting $10M in exchange for some amount of personal stock.