While games are stored in the game explorer, as a free member you will not be able to view them.
What you should do, is go to the bottom of your game page, where the game will remain for the next month or so. Click on it, and you can go through the game. The chatbox also functions as normal, and your opponent will get an alert whenever you have typed something in it.
I played my first two games on the site yesterday, seems a very nice place. After the game, my opponent and I wanted to go back through the game and assess what went well/not. Not that either of us are what you'd call 'good', but anyway, I clicked on 'game explorer', thinking that'd be where i'd find my games, and it said 'the player was not found:'. Are games not stored for ordinary members, or am i doing something daft and not clicking the right menu :)
Thanks in advance