I think there are two big reasons why they would do this: 1. Investment They will likely make a multiple of the $25M when Square IPO's. The best possible evidence of Square's traction is what Starbucks just provided. Smart investment 2. Focus on core competency World class payment processing is not a significant source of competitive advantage for Starbucks, but it is for Square. This deal allows Square to worry about payment acceptance & processing efficiency, and Starbucks to focus on product/service. Michael Porter would be proud of this deal.
But I don't mean to imply that's the only factor. I agree about the investment.
Moreover there's the customer experience of paying with Square. It's nice and you don't have to set up a Starbucks card and put money on it. You just pay. It delights customers and may move lines faster, but perhaps most significantly, it lets Starbucks track loyalty. The business value of that is tremendous.
Starbucks basically just upgraded their app in a big way. Remember they were the earliest pioneers of "pay with your iPhone" with ther Starbucks app.
It doesn't hurt that Square is associated with small scale "craft" businesses either. Starbucks is trying to revitalize their brand right now.