Most cable providers authenticate their users by checking the hardware address of the network card connected to the modem. When you connect your AirPort Extreme base to the modem it sees a different network card than the one it wants to see, so it refuses to connect.
You can do several things. Before you decide, read your user agreement carefully-your cable provider may specifically prohibit connecting more than one computer to their equipment! They assume that any wireless device is going to connect to multiple computers so they disallow them immediately.
What I would do is spend $30-$50 and buy an inexpensive gateway router (also known as cable/DSL router). This goes between the modem and your computers and ABS. The modem sees the router, not any of your computers, and most can be configured to masquerade as your original computer's network card (that's called "MAC address cloning") so the modem doesn't even notice a change.
Another option is to call the cable company's tech support and ask them to add another MAC address to the allowed list for your connection. It may work, or they may absolutely refuse. They may watch your connection closely after such a call. Ok, I'm paranoid, but I don't believe in giving them a reason to look.