I can't add an external tester in TestFlight because the Add button is greyed out I'm trying to add an external tester in TestFlight, but the Add button is greyed out:



Is this because the app hasn't yet been approved by Apple, or some other reasons?

I tried in both Chrome 58.0.3029.110 (64-bit) and Safari 10.1.1 (12603.2.4) on macOS 10.12.5.

This same person is beta-testing another of my apps, so I'm sure the problem is not regarding their account or mine, just the particular app I'm trying to add them as a tester to.
You pasted the email address in the "Last Name" field. Simply move the mail to the first textbox with the "Email" label above.