How does an app enable transfers between two different bank accounts?

Depends where in the world you are based. Assuming you are in the US, then the most common way is to use an ACH (ACH) methodology to transfer.

• To do so, you can hook with any ACH provider (there are 100s of them out there, if not 1000s).
• You do not need a licence if you are doing it for your own company and are pushing payments out to your employees.
• You do not need permission from your end-account holders either.
• You would require their bank account information, name, etc.
• You would need to authorise the ACH provider to debit your account and push the payments out
• You simply create a text file (in most cases) - commonly known as an ACH file and send it out to your ACH processor for further processing.
• The file will contain itemised breakdown as per the given format by the ACH provider for each employee you wish to pay and the amount, etc.
• Some ACH provider now give access to their APIs. So you can program your app to take advantage of the API and push the payments out.
• You do not need to integrate with any bank directly. Just the ACH processor.