What does "I'm game" mean and what is its correct usage?
I'm game means that
I am up for the challenge, but arguably more commonly,
Q: Fancy going for a drink?
A: Sure, I'm game...
Q: I was thinking of going shopping. Do you want to come?
A: Yes, I'm game...
As for correct usage, as with most idioms, use it as you will... there is no correct usage as such.
I'm afraid to go down the path of explaining correct usage of an idiom, so I will just tell you how I use it! I think of it as another way of saying, "That sounds good to me." Another phrase people often use is, "I'm down," and I would say that these two can be used interchangeably. Here is an example:
Q: Do you want to go see a movie tonight?
Possible replies all meaning the same thing:
A: Sure, I'm game.
A: Sure, I'm down.
A: Sure, that sounds good to me.
That last one is a mouthful, so you can see the appeal of the first two!