How does rental car insurance offered by credit card companies (when you use the card) work?

Question by hedonist: How does rental car insurance offered by credit card companies (when you use the card) work?
I have credit cards which say they cover me for rental car insurance when I use the card to make the full payment for the rental. How does that work? Are there any catches? Does it make sense for me to get the additional insurance from the rental agency (like Avis)? If I am in an accident, will the Credit Card insurance cover everything or will it also be charged to my regular

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