As others have said, open communication is critical. It is necessary but not sufficient.

You've probably been thinking through why you cheated and continued to cheat. However it can be really difficult to go deep get the true answer by yourself - brains tend to generate reasons/excuses in a way that minimizes your responsibility and preserves your ego as best it can. If you try to explain what happened to your wife and give a facile or self-serving excuse, you could make things far far worse.

Many people find that the process of talking with a professional (a counsellor or therapist) can get deeper than doing this by yourself. You will get to a more profound and authentic understanding of yourself and of steps you can take to be the better person you want to become. By knowing yourself better you are able to properly apologise and explain to your wife why you betrayed her trust. You will also be able provide some evidence that you are not going to do this again. Broken trust takes a long time to repair - self discovery and improvement is a process, not a single event.

Another thing to consider is whether you and your wife can have constructive conversations about what happened and what your hopes and wants are for the future. If conversations rapidly devolve into arguments and anger, it may make things worse (but every relationship is different). If you worry that those conversations may spiral out of control, or will not be productive, I'd suggest doing this with a neutral, professional third party like a relationship councillor who can facilitate the conversation.

Those are a few ideas - they are certainly not comprehensive and YMMV.