How would a true push (and not poll) method work, if all you have is the local thunderbird client? If I had to guess, the question from the other person is not related to mail poll/push stuff
Usually you rely on a server telling the mobile app that there's something new to poll (or the message is directly sent and displayed).
A TCP connection to the IMAP server is kept open, with the server sending a packet whenever anything interesting happens (see IMAP IDLE), waking up the client app. This is how all push notifications work nowadays, IIRC, since you can't send information directly to clients without a persistent connection, due to NAT.
Usually you rely on a server telling the mobile app that there's something new to poll (or the message is directly sent and displayed).