This sort of thing happens regularly, and most credit card companies will do exactly what yours is doing. You are not responsible for tracking down the person who is using your credit card. In many cases, the company that accepted the card without adequately checking ID is on the hook for the amount of the purchase, but that's a risk they've accepted because, for example, they make more money by having an easy, streamlined checkout procedure than they lose from taking occasional stolen cards.
I don't know how vigorously credit card companies go after people using stolen cards. It seems likely that there is enough information for them to do a lot on that front: if the thief has ordered something to be shipped to a particular address, the credit card company and/or the police could go to that address and arrest the person who accepts the package. (Of course, somebody has to notice the fraudulent charge before the package is delivered!)
All you can really do to prevent this in the future is keep an eye on your credit card bills for purchases that you didn't make, and be careful not to give out your personal information unintentionally. Even so, it's not terribly difficult for thieves to obtain credit card numbers and make purchases. It's unlikely that you were targeted personally; probably, you or your credit card number were just in the wrong place at the wrong time.