An Albany, Georgia woman is fortunately recovering from necrotizing fasciitis (flesh eating disease) that developed after a cat bite. The cat presumably wasn't the source of the bacterium that caused the disease, but it was the source of the wound that let the bacterium (usually Group A Streptococcus sp) get into her body in the first place, and from there the infection spread rapidly.
In this case, the woman was bitten on the hand by her cat. There's no mention of what she did after the bite, but often people don't take adequate precautions after cat bites because they can appear minor. While the trauma can be minor, cat bites are notorious for causing infections because they can drive bacteria deep into tissues, making it easy for infections to develop.
Four days after the cat bite, the woman's hand "blew up" and she was rushed into surgery. It's an emergency situation because death rates can be high, and those people lucky enough to survive can still have serious complications sometimes requiring measures as drastic as limb amputation, and the extensive tissue damage can lead to chronic problems for the rest of their lives.
People shouldn't worry about getting flesh eating disease from their cat (in part because people don't actually get this kind of infection from cats, it's just that bites can predisposed them to this (and other) kind of infection). However, people should be aware of potential issues associated with bites (from cats in particular, but also from any kind of animal). This case is just one more warning about a range of potential problems that can develop following a cat bite. Any cat bite should be taken seriously and at a minimum promptly and thoroughly cleaned. Bites that occur over the hands, feet, face, joints and similar high risk sites or that occur in people with compromised immune systems should be evaluated by a physician - and sooner rather than later. It's much easier and better to prevent an infection from developing than to try to control an established infection, and while most cat bite infections are mild, serious problems are far from rare.
More information about cat bites can be found on the Worms & Germs Resources - Pets page.