In this answer, I won't elaborate on the possibilities of fraud (or pure human error), because something can always go wrong. I will, however, explain why I think you should always keep receipts.
When the (monthly or so) time comes to pay your credit card bill, your credit card company sends you a list of transactions. That list has two primary purposes, both of which I would consider equally important:
- Showing (partially censored) copies of the list of transactions to third parties (e.g. your employer) in order to prove that you paid for something so you can get a reimbursement.
- Checking for yourself whether the single sums in the list are correct.
While for the former item, a receipt is not necessary (though it certainly does not hurt showing the receipt along with the bill to provide further proof that the payment was indeed connected to that bill), the latter point does require you to store the receipts so you can check, item-by-item, whether each of the sums is correct (and matched with a receipt at all). So, unless you can actually memorize all the credit card transactions you did throughout the past one or two months, the receipts are the most convenient way of keeping that information until the bill arrives.
Yes, your credit card company probably has some safeguards in place to reveal fraud, which might kick in in time (the criteria are mostly heuristical, it seems, with credit cards or legitimate transactions here getting blocked every now and then simply because some travelling of the actual owner was misinterpreted as theft). However, it is your money, it is your responsibility to discover any issues with the bill, just as you would check the monthly transaction list from your bank account line by line. Ultimately, that is why you sign the vendor copy of the receipt when buying something offline; if you discover an issue in your list of transactions, you have to notify your credit card company that you dispute one of the charges, and then the charging vendor has to show that they have your signature for the respective transaction.
So, to summarize: Do keep your receipts, use them to check the list of transactions before paying your credit card bill.
EDIT: The receipt often cannot be replaced with the bill from the vendor. The bill is useful for seeing how the sum charged by the respective vendor was created, but in turn, such bills often do not contain any payment information, or (when payment was concluded before the bill was printed, as sometimes happens in pre-paid scenarios such as hotel booking) nondescript remarks such as "- PAYMENT RECEIVED -", without any further indication of which one of your credit cards, debit cards, bank accounts, stored value cards, or cash was used.