This thread was extremely helpful -- thanks to all who provided insights. I was having this same problem trying to access my camera feed through my iPhone 6. But I got it working tonight.
Here's basically what you have to do, if you're using a Mac (my network is Comcast modem to Airport Extreme for WiFi):
1) Set a static IP address for your Mac -- follow these instructions:
http://www.macinstruct.com/node/550 -- let's call the static IP address: "10.0.1.130"
2) Hold on to that static IP address and then go to Airport Utility and select NETWORK, and then create a new Port entry. In the Description field, selected "Remote Login" from the dropdown menu.
3) Add the Port number (8848) to the Public and Private TCP Port fields *only*.
4) In the "Private IP Address" field, make sure you change the numbers to your static IP address: "10.0.1.130" Then hit "SAVE".
At this point, if you go back into Sighthound Preferences on your computer, the "External Address" will probably still say "Attempting to Connect..." But don't worry. Sighthound warned us this would happen.
5) Now open up the Sighthound iPhone app and in the first line, you have to type in your home network's "External IP Address"
Here is how you get the External IP address (as Ryan mentioned): go here:
http://www.whatsmyip.org) and then plug that into the first line of the mobile app login screen, followed by the Port address (8848). So example: 74.213.239.105:8848
6) Next, type in the User ID and Login Password that you initially configured in the Sighthound app.
After constant testing and trouble-shooting, this worked for me.
I think Sighthound would do everyone a favor if they did a walkthrough blog post (with screenshots) of how to connect the iPhone app to the external IP address using an Airport Extreme. A lot of users (like me) bought the $60 software on the assumption I wouldn't be spending 3 days trying to get the app to work. I know you don't want to have to do these types of walkthroughs for all kinds of routers, but I'd imagine Sighthound is getting a ton of new Apple users who are accustomed to easy setup. You don't want to lose them.
EDIT: I just discovered that in order to be able to access the Sighthound video from the iPhone, your computer running the Sighthound software must be at least actively connected to the Internet and not in sleep mode. This seems to be the case when using the iPhone app to connect via the "internal" and "external" IP addresses. Kind-of a bummer.