Probably one of the clearest concise explanations of what having a relationship with God is like, from the book, Bono on Bono: Conversations with Mishka Assayas.
Occasionally [Dad] would ask me a real question, meaning I had to give him a real answer. It was always about my belief in God: “There’s one thing I envy of you. I don’t envy anything else,” he said to me one time. But think about it: I was singing, doing all the things he would have loved to have done, had a creative life. He said: “You do seem to have a relationship with God” And I said: “Didn’t you ever have one?” He said “No.” And I said: “But you have been a Catholic for most of your life,” – “yeah, lots of people are Catholic. It was a one-way conversation… You seem to hear something back from the silence!” I said: “That’s true, I do.” And he said “How do you feel it?” I said: “I hear it in some sort of instinctive way, I feel a response to a prayer, or I feel led in a direction. Or if I’m studying the Scriptures, they becom alive in an odd way, and they make sense to the moment I’m in, they’re no longer a historical document.” He was mind-blown by this.