How does a USA chess player get invited to a chess tournament like the China-USA Chess Summit (2013)? How does a USA player get invited to such a tournament if it were to happen again? If I were to make a goal to play such a tournament what steps would I have to take? Just curious on how the registration process works since tournaments involving more than one nation seem to be closed compared to regular tournaments in the US. It would be really helpful if someone could explain how you register or get invited for either country involved.
The participants, at least on the US side, were roughly the level of players that are generally expected to play in the US Championship, although the top few players (at the time, Nakamura, Kamsky and Onischuk for the men and Krush, Zatonskih for the women) did not take part. That would imply that if you're male, you would need to be a solid grandmaster, and if you're female, probably 2200 FIDE at least. My hypothesis is that the USCF went down the list of US ratings from the top until they found enough willing players to fill their roster. 

Becoming a GM is extremely difficult and you need more motivation than being able to play in a single tournament. If you're female, 2200 FIDE (about) is easier but will still require years of effort depending on your current skill level.