Gene want to tell Finny the truth about what happened and begins to do so. Why is it important for him to tell the truth?

Gene had to. The guilt was eating Gene up. The truth was always swept under the carpet, so to speak, and replaced with cryptic suggestions about what might have happened. Gene's guilt was obviously too much. More than that, Gene felt an open confession, without any excuses, was needed for some kind of cleansing. As long as a full confession was not made, there was toxicity in their relationship. It prevented the boys from becoming each other (I know that sounds strange but it is one of the themes of this novel.)