Their venom causes necrosis but the real scary part is that it never stops. Venoms are normally broken down by the body or just use themselves up causing their damage, however brown recluse venom works forever. It just keeps on spreading and killing flesh around the bite until either a doctor cuts a big enough chunk out of you to stop it or secondary infection kills you.
This is not actually true. The wounds take a long time to heal, and they sometimes debride the necrotic tissue, but that's to prevent infection if it starts to get a bit rotten. However, they certainly don't cut the venom out, or anything. They actually prefer to leave the necrotic tissue in place, to protect the wound and aid in healing. If you have to debride it all, especially in a deep wound, where it would be more necessary, it will leave a huge open hole that will take forever to heal properly, and will tend to scar badly.
Thankfully recluses are very timid and non aggressive (hence the name), and the vast majority of bites do not become necrotic, and even the majority of necrotic bites heal alright. However, in rare cases the necrosis can become systemic (more common in fat people, since it destroys fat tissue preferentially), and can start breaking down tissue all throughout your body. Understandably, this is not much fun. See
wikipedia's take on it for some good information and gory photos.
There are some interesting anticoagulant colubrid snake venoms that can last a long ass time. There's a japanese snake, the Yamakagashi, that can cause lethal internal bleeding several months after a bite. It's an interesting snake because it eats toads and sequesters their poison, making it one of the few snakes that is both venomous and poisonous (garter snakes that eat highly toxic newts can be poisonous as well, using tetrodotoxin, a really cool paralytic neurotoxin usually found in marine organisms, like pufferfish (deadly sushi!) and blue ringed octopus. Garter snakes are also mildly venomous, though I don't think they use TTX in their venom). I'm pretty sure I read about an african bird snake (
Thelotornis) causing serious clotting problems up to several years after the bite, but I can't find my reference for that, so I might be making that up.