No, Facebook does not have access to your bank details or your bank transactions.
When you sign up, your account setup (including linking your bank account) is done in a secure webview. A webview is a website embedded into Facebook Messenger and it is generated by our servers, not by Facebook. Your bank login is transmitted and encrypted from the webview to our server and is then forwarded to Yodlee, our aggregation partner, for further encryption and safe storage. We don't store your bank details.