I started the game last night and had to stop a few minutes in. I came back this morning...so 12 hours of inactivity. Obviously, the game lost its connection to the server when I went idle. I got a dialog as such and it started over with the Activision logo and the intro sequence. However, there was no audio whatsoever. I had to start another game on the Xbox and then restart Destiny to get the audio to come back.
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I've found this with other games as well. Its a side effect of the multitasking functionality of the OS. after you're finished playing, hit the guide button to go home, then while highlighting the preview tile, press menu (start) and choose the quit option. This will reset the program so it loss properly next time and will avoid this problem.