http://iscoresports.com/

What am I missing here?

There is no "score a game" button to get started with.

I have "team manager" and "game manager" but I don't see anything to take me to actually start scoring a game.

Go to Game Manager, and click the "New Game" button. Starting with the iPad release, we got rid of "Score a Game" as it was really redundant --- games now work like the rest of the system where you go to the manager, then add a new one.