If we hear a preacher or someone witnessing for Christ who tells us we need God, our response might be, "for what?" Believe it or not, it is even possible for someone to self-identify as a Christian and slip into the same stream. Oh, these are not antagonistic toward God, or anything, it's just that they (even if they would never admit it) think God would be proud of them. If someone needs help, they figure God needs theirs.
When people are self-satisfied, they feel no need for God.
When our thoughts are invaded by the stupid, the silly, the sinful, or the absolutely debauched, a hint arises within us that maybe we're not quite so altogether as we had assumed. There's something in us we don't quite understand, itching to make us blunder and look all too much like the rabble of the unwashed masses. It's not so easily put under reins either. Maybe we don't have this life thing mastered.
It is the imposition of an unwanted thought, an undesired desire that arises within us embarrassing us as it sprouts into consciousness, which breaks the illusion of our self-control. Perhaps it is an irrational fear, or a secret prejudice that shakes our self-reliance. At some point, the failure that such things inspire breaks out in the open. In those moments we discover that we can't do this life thing by ourselves after all. We're less than we thought we were, we do need help, and from someone greater than ourselves--we need God.