How can I gain my wife's trust back after cheating?

Describe why you had the affair, without justifying it, and without blaming your partner. Get the help of a therapist for this. The scariest thing your partner can hear you say is that you don’t know why the affair happened, because then why wouldn’t it happen again? This will cause your partner to have constant anxiety. You should also be consistently transparent from now on and going forward. Your partner is not listening to what you say, they are watching how consistently transparent you are because that communicates to them how serious you are and your level of commitment to the relationship. Know that it will take time for your partner to trust you. Let them know that you understand that earning back your trust will take time and consistency.