Your firewall/NAT/router thing takes the one IP address given to you by your ISP and distributes it amongst the computers at your house. It does this by taking that IP address for itself and handing out totally different addresses to your computers that the internet at large doesn't know about. When those computers try to make a connection to the internet, it rewrites the envelopes of the data so that the internet sees that data coming from the ISP address so that the computers on the other end know where to send a response back to. The firewall thing then sees the response and can associate it with the computer that sent out the original request and rewrite the envelope so that it gets it.

Where this breaks down is when a computer on the internet wants to send data directly to one of your computers. There's no association to be made for the firewall to know where to send the data, so it just drops it. This is part of its firewall-ness. You can configure it so that it knows that certain types of data go to a specific computer, but this fails when you don't know which computer that's going to be.

AOL's ancillary AIM protocols work a little backwards. Your computer sends a message to the remote system to tell it to start up a new connection back to your computer. When it does so, the firewall sees a new connection, but has no way to associate that with the original AIM communication, doesn't know what to do with it, and drops it on the floor.

It would be possible for a firewall to inspect AIM communications so that it could make an association for the incoming communication, but I don't know of any that actually do this. The other option is to configure your firewall to forward all such incoming transmissions to go to a specific computer, but that means that that sort of communication will go only to that computer. In other words, if you wanted to do an AIM file transfer, you have to be on that computer.

This explanation avoids a lot of the technicals, but I think that it's not inaccurate, just imprecise.