The reason you fall asleep faster with the TV on is likely because of the sound which helps shift your mind from other things.
When I sleep without the TV on my mind is usually racing with other thoughts and when I have the TV on the sounds of the TV change my thoughts.
So I'm thinking more of nicer thoughts on the TV than what would be going through my head.
That could be a reason why you sleep better with the TV on.
Some people sleep better in complete silence but some people sleep better with noise in the background.
I always sleep with the TV on now on low sound and I fall asleep faster and get a more restful sleep due to the noise.
I think more when I'm in complete silence which keeps me awake but when the TV is on my mind is basically shutting down to help me sleep.
The sound of the TV can also be soothing to some people.
Even listening to the radio on low can help some people fall asleep.
Some people just need white noise to help them fall asleep and stay asleep.