When I quit drinking 14 years ago my kids were 5 and 7. They had seen me drunk, but they didn't know I was drunk: they only knew that I was mean to them sometimes and that I'd been carried off to the hospital once by paramedics.
I too struggled with what to tell them. Addiction is hard enough for US to understand, much less for little kids. I decided the best thing was to explain to them what they could understand and then expand on the explanation as they got older.
So at first I told them that mommy had been sick and the sickness had made her mean. I apologized and said I was working on getting better and being a better mommy. As they got older I explained more and more including the possibility that they may be predisposed themselves to become addicted. I never hid anything from them--but I did consider their ages when deciding what to say.
Now that they are in college, they are faced with their own decisions about drinking and drug use and the cool thing is that they talk to me about such things very comfortably, because we've established that over the years.