My take on it is that it helps us addicts focus on the here and now. Rather than living in the future. Or in the past.
Here and now is a place where things are manageable. I can get up today to go to work, even if I'm not sure I could do it the rest of my life. The pain in my back might be severe today. But is it going to stay with me from now on? Today I may be broke, or not have a job. But will this always be the case?
If I stay in the present, I face life in manageable chunks. If I speculate about a problematic future, I might become overwhelmed to the point of picking up a drink or drug.
If I'm managing an archaeological expedition through the wreckage of my years of drinking and drugging and being irresponsible and abusing I might get lost and never get back to today. Living in today, I don't need to open the door and enter that tunnel to my dark past. Instead, I stay in today, where where the light of the moment shines upon my activities and keeps me focused on what's real.
The idea of living in today did not originate with the framers of the 12-step programs. Eastern religions for centuries have taught the value of focusing upon this minute, this moment, this second that God has given us.
Regardless of where it came from, the concept of living a day at a time teaches us that life is manageable - something each of us in recovery can use to our benefit.