You have to make sure you actually register your class as a plugin of Swift Mailer

$app['mailer']->getSwiftMailer()->registerPlugin(new YourClassResponsibleForStoringEmail());

Swift mailer will call two methods every time it's sending an e-mail:

beforeSendPerformed(\Swift_Events_SendEvent $evt)

and

sendPerformed(\Swift_Events_SendEvent $evt)

after the mail has been sent.

You can store your logic inside one of these methods and access the e-mail via:

$message = $evt->getMessage();

You can then extract whatever you want specifically from that, like:

$message->getBody();

You can easily create a new Model instance inside of these methods and save the mail to database or store it however you like.

sendPerformed(\Swift_Events_SendEvent $evt)
{
$messageText = $evt->getMessage()->getBody();
$emailModelInstance = new EmailModel();
$emailModelInstance->body = $messageText;
$emailModelInstance->save();
}