RabbitMQ is an open source message broker software written in Erlang. It uses exchanges to route messages from producers to queues based on routing keys or bindings. There are four main exchange types - direct, fanout, topic, and headers. Mule connects to RabbitMQ using the AMQP connector. Flows in Mule can send messages to queues via exchanges and receive messages from queues via exchanges, as demonstrated using a direct exchange to send a message to a queue and have another flow receive it.