To build a RailsGame, first you'll need an idea for a game. Think about browser games you've seen, and browser games you haven't. Remember that whatever your idea is, you'll need to find, make or buy all the pictures, text or animations you're thinking of! Find a friend who's an artist, learn to draw or write, or get ready to spend a bit of money...
So let's say you've got that angle covered. Think of your basic game design. Now, think of the smallest piece of it that will make an interesting game. Something you think you could put together in a week or two (or less!), just to have something to show off. That's where you should start.
Now, divide your idea up into things that look like "normal" web pages and parts of the game. For instance, you probably want the player to log in on a pretty standard-looking web page. If you're writing a game like "Master of Orion" or "Civilization", there will be a lot of pages that sit and wait for instructions -- think of the screen with all the cities on it, for instance.
For the normal web pages, you can build them in Ruby on Rails, just as you always would. After you've created your RailsGame directory, you could even put one or two together, just to get started.
For pages that breathe, though, you'll wind up adding some RailsGame containers. Containers are 'windows' in the user's browser for things like player-to-player chat, or pictures of who is nearby, or the section of a game map that the player can see.
It's possible to have parts of the web page that aren't containers, of course. For instance, a help system frame displaying HTML help wouldn't need to talk to the RailsGame server, so it could exist in the page without any RailsGame settings at all. It could be a standard IFRAME with nothing special, and nothing connecting it to the server.
Where RailsGame becomes interesting is when you need to send something back to the client. That is, when the person at the browser takes no action, but still sees a result. In chat, that would be when another person has chatted to them, or when the map changes, or when the game window above says something like "Bob has entered the room". To enable that, you either need AJAX polling (inefficient and not very reliable) or server push. RailsGame exists specifically to enable server push.