Start with the truth, nothing but the truth, and then keep being honest. Even if your spouse is focussing in on the details of your affair, what he/she really wants more than information is assurance that you can be trusted.

Your #1 job for a while will be this: Create a safe place for your spouse. Do whatever is necessary to assure your spouse that they do not need to continue fearing betrayal. If you really want to make a difference, don't wait for your spouse to tell you what they need to feel safe; take the initiative to do things on your own (access to things that were previously private, inclusion into more areas of your life, open communication, frequent verbal assurances from you, etc.) and then ask what else would help them feel safe. In time (plan on months, not weeks), the insecurity will diminish and things will begin to feel more "normal" again.

By the way, the best motivation for making these kinds of changes comes from a desire within you to be a different kind of person: honest and trustworthy (an internal shift, in your control). When you are focused on who you are becoming, you can continue to make those changes even if you don't get the response you want from your spouse. But if you change simply in order to make your spouse happy (an external shift, out of your control), you'll likely get discouraged rather quickly with your spouse's anger and/or suspicion. Then it becomes easy to give up trying and blame your spouse for not being able to "get over it."