Your Airport hopefully has DHCP on - this hands out dynamic IP addresses in your home network to all computers connected wirelessly or not.
Most internet providers only feed you a single IP address for the internet. That's why you need a router - to split that IP up. To the ISP, even if you have 1 or 10 computers, it all looks like just one is connected.
So what I'm saying, and others are too, is that since your Airport only has one LAN hole, by using a switch, you are creating a daisy chain. This way you have expanded your Airport to 5 or more LAN holes. Now your Airport will be able to give out 5 local dynamic IPs.
Get it?