It's been probably a decade or more since I read it, but I remember it having a similar effect on me at the time. I don't remember what specifically in there had that effect anymore though, so I'm probably due for a reread.
If I remember correctly, it was about Tolstoy's struggle to find meaning in his life, even after having a wife, several kids, and finding lots of success with his writing and a large house. He experiments with and discusses quite a few philosophies in a fairly frank manner before circling back around to Christianity, and does his best to make a case for it based on logic and comparing with his past experiences.
But even for people who aren't religious (I'm not really that religious myself) and don't want to be, I think they'll still relate to his struggle to find meaning and find some useful and thought provoking ideas in there.
I do agree that more (most?) people should read it. I even gifted a copy of it to a good friend at the time.
Other books where I had a similar experience from reading them include Siddhartha by Herman Hesse, The Conquest of Happiness by Bertrand Russell, and The Last Lecture by Randy Pausch.
In particular Siddhartha had a lot of similarities to A Confession, in subject matter and structure, although it's told as a third person narrative and not as a personal account.
What were some of the takeaways you got from reading it?