What does the Broken pipe message mean in an SSH session?

It's possible that your server closes connections that are idle for too long. You can update either your client (ServerAliveInterval) or your server (ClientAliveInterval)

ServerAliveInterval
         Sets a timeout interval in seconds after which if no data has
         been received from the server, ssh(1) will send a message through
         the encrypted channel to request a response from the server.  The
         default is 0, indicating that these messages will not be sent to
         the server.  This option applies to protocol version 2 only.

 ClientAliveInterval
         Sets a timeout interval in seconds after which if no data has
         been received from the client, sshd(8) will send a message
         through the encrypted channel to request a response from the
         client.  The default is 0, indicating that these messages will
         not be sent to the client.  This option applies to protocol
         version 2 only.

To update your server (and restart your sshd)

echo "ClientAliveInterval 60" | sudo tee -a /etc/ssh/sshd_config

Or client-side:

echo "ServerAliveInterval 60" >> ~/.ssh/config