Answer by replying to the email
Every notification is replyable, so a conversation can be answered from your mail client.
When teem emails you about a conversation, you can answer it by replying to that email. Your reply lands on the conversation and reaches the person in their widget, exactly as if you had typed it in the inbox.
The same is true in the other direction. When someone gets an email saying you replied, their reply to it comes back into the conversation.
Why the address looks strange
The reply-to address contains a long token. That token is the only thing that decides which conversation your message joins, which is why it has to be there and why you should not edit it. A From header can be forged; the token cannot be guessed.
What teem ignores
Out-of-office replies and other automated mail are recognised and dropped rather than posted into the thread. An empty reply is also dropped, so a stray send does not produce a blank message.
If a conversation had been resolved, a reply reopens it and it returns to the inbox.
If a reply does not appear
Anything teem cannot place is quietly accepted and dropped rather than bounced, because a bounce would tell a stranger whether a conversation exists. That means a misrouted reply looks like silence.
Check that you replied to the address teem sent from, with the token intact, and that your mail client did not strip it in favour of a friendlier one. If replies stop arriving across the board rather than for one thread, it is the mail route rather than the message.