If you horn in on a conversation you are interrupting a conversation. Another way of saying to “horn in” is to “butt in”. Both are informal. Here the Rhinoceros is literally horning in on the conversation because he has a horn (the sharp thing on the end of his nose).
1. If I hear a tourist trying to speak Portuguese or Spanish and they are having difficulty communicating with the locals, sometimes I horn in on the conversation and help them.
2. If a tourist is paying too much for a souvenir, I might horn in on the conversation/ negotiation and tell them in English that they can definitely get a better price.
3. When you disagree with someone’s opinion, especially in a group of friends, it’s not a big deal to horn in on the conversation and give your 5 cents (give your opinion).