Taking ownership of the choices to cheat which resulted in the lower grade, and whatever reasons you earned the "F" is what they are looking for. Not saying, "I was caught cheating" this (even if not your intent) would read by an admissions counselor, "Well, if I hadn't gotten caught, I am okay with cheating."

Just state the facts, don't make excuses, don't blame shift, don't attempt to minimize, don't go over the top with "I felt so ashamed" (this could read manipulative even if not intended) and be sure to clearly state how this mistake you made you have grown, learned, and how you clearly have made effort to change etc. If you are unsure if thats what you are doing, have someone you trust who can be ruthlessly honest with you to proofread your addenda.

Remember that feeling guilt is good if you made a mistake, it can lead to improved character and growth. But shame will keep you stuck. May want to listen to Brene Brown & Adam Grant's podcast "The Curiosity Shop" they have numerous episodes on apologies, shame, taking ownership etc that may be helpful. This is also excellent: https://www.psychologytoday.com/us/blog/brainstorm/202603/guilt-and-shame-are-not-the-same

Lawyers are held to an extremely high standard of ethics, morals, and law abiding. Being admitted to the Bar is much harder than to law school if there is dishonesty, excuse making, character issues not addressed but discovered later.

If there are extenuating circumstances for a low grade, such as a death in the family, admissions committees do want to know this. But, don't write tons or try to get overly emotional.

From having spent almost my entire career working in top university ethics departments, this is the advice I would give. I hope these are helpful, and I wish you the best! Don't lose hope, I heard a podcast from a consulting firm where they talked about character/fitness issues and discussed a very highly respected law professor who had served prison time, changed his life around and became a phenomenal lawyer (and obv. law prof) later in life.