The application sends emails using the Twilio SendGrid API with Action Mailer and updates their status based on the event webhooks sent by SendGrid. To build this, one needs Ruby, Bundler, ngrok, a Twilio SendGrid account, and to set up an environment with Rails. The application is configured to use the sendgrid-actionmailer gem for sending emails via the SendGrid API. An email template is created within the application for sending emails too. A controller is added to receive webhooks from SendGrid and update the Message object based on the custom parameter sent in the event. A route is also set up for this action, and ngrok is used to tunnel the webhook requests to the local machine. The application is tested by sending a message, waiting for the event notification, and updating the status of the email in the list.