The network connection only needs to exist while messages are being exchanged, not while the applications are running. When a message reaches its final destination queue, it can be read and processed by the application serving that queue. The model used within message queuing systems can be thought of as a generalization of the technology underpinning e-mail. Systems that provide e-mail rely on the ability to queue messages at servers until users are ready to receive them. Message queuing systems are just the same but store messages for applications rather than people.
Queues hold messages in transit or while the receiving applications are unavailable.
A key difference between e-mail and message queuing is that senders and recipients are applications, not people. Loss of an e-mail is potentially serious. Loss of a message between two applications can also be serious. However, this time there is no guarantee of independent follow up. The lost message might represent some internal communication deep within an application.