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When this question was first asked, Robomongo did not support the MongoDB 3.x series. The good news is that support has been added more recently. Check out their blog post on the topic.
More generally for users of the DigitalOcean MongoDB One-Click, you can find more information about how it is configured in this tutorial:
By default, connections are restricted to localhost. In order to enable access over the internet, modify the value of bind_ip in /etc/mongod.conf Before doing so, make sure to check out the security checklist from the MongoDB documentation.
Robomongo RC8 (release notes) recently added support for making connections via an SSH tunnel. This allows for leaving MongoDB bound to localhost while still being able to make a secure remote connection.