In addition to the general exposure, a company like Citi gets its foot in the door to handle all of the banking for the team and the venue, access to players and their agents and a place to drum up business with other team sponsors in the corporate suites.
Getting users to sync their credit cards is critical to the proving-it part, allowing Foursquare access to detailed purchase data that bolsters its claims to be an effective tool for vendors trying to drum up local busines.