LWT messages are not really concerned about detecting whether a client has gone offline or not (that task is handled by keepAlive messages).
LWT messages are about what happens after the client has gone offline.
The analogy is that of a real last will:
If a person dies, she can formulate a testament, in which she declares what actions should be taken after she has passed away. An executor will heed those wishes and execute them on her behalf.
The analogy in the MQTT world is that a client can formulate a testament, in which it declares what message should be sent on it’s behalf by the broker, after it has gone offline.
A fictitious example:
I have a sensor, which sends crucial data, but very infrequently.
It has formulated a last will statement in the form of [topic: ‘/node/gone-offline’, message: ‘:id’], with :id being a unique id for the sensor. I also have a emergency-subscriber for the topic ‘node/gone-offline’, which will send a SMS to my phone every time a message is published on that channel.
During normal operation, the sensor will keep the connection to the MQTT-broker open by sending periodic keepAlive messages interspersed with the actual sensor readings. If the sensor goes offline, the connection to the broker will time out, due to the lack of keepAlives.
This is where LWT comes in: If no LWT is specified, the broker doesn’t care and just closes the connection. In our case however, the broker will execute the sensor’s last will and publish the LWT-message ‘/node/gone-offline: :id’. The message will then be consumed to my emergency-subscriber and I will be notified of the sensor’s ID via SMS so that I can check up on what’s going on.
In short:
Instead of just closing the connection after a client has gone offline, LWT messages can be leveraged to define a message to be published by the broker on behalf of the client, since the client is offline and cannot publish anymore.