This Tutorial Shows how to setup GrooveMail to use SendGrid SMTP for Email Sending, Including how to Verify Your GrooveMail Sender
GrooveMail recently changed to requiring an external SMTP service to send the emails.
Think of GrooveMail as the Post Office, where you manage your mail options (Lists, Campaigns, etc) and SMTP as the Mailman that does the actual delivery.
Previously GrooveMail handled this all, but due to issues they recently changed to requiring a separate SMTP provider.
This tutorial shows how to configure Sendgrid (https://sendgrid.com) as your SMTP provider. Sendgrid has a free account that allows you to send up to 100 emails per day, and lower priced plans for more sending as needed.
Groove.cm plans to re-introduce GrooveMail SMTP in the future, as well as others such as Mailgun, Amazon, etc.