It means "my alarm clock did not buzz/ring/make a loud noise to wake me up this morning." For example, you can say: I was late to work today because my clock didn't go off, so I overslept.

"go off" means it's opposite "turn on" and in this case his 'clock' is an alarm clock. So, the alarm clock did not turn on this morning. Either it is broke, he forgot to set it, or he set it for the wrong time. Hope this helps!