You will need to install an MTA (Mail Transfer Agent) on your droplet in order to route mail for PHP. You can do this simply by installing postfix or sendmail on your droplet.
sudo apt-get install sendmail
Configure hosts file correctly: nano /etc/hosts And make sure the line looks like this: 127.0.0.1 localhost localhost.localdomain yourhostnamehere
Run the sendmail config and answer ‘Y’ to everything