From what I know of the American market, this is a bigger innovation there than it'll be here where emailing money is fairly common (at least with young people). For the moment it's US only anyway.
I think this is a clever move by Square because it helps move them towards debit access. It's the dream of all online payment companies to move away from the credit card system (controlled by two companies with high fees).
I haven't looked into this further but the thought on Hacker News is that this is being done by having accounts at all the major banks and then transferring money between the sender's account and the Square account at that bank and then from the Square account at the other bank to the recipient's account.