The MapBox street style is private, you definitely can't copy it from anywhere.
You can archive similar style ( except for the lines connecting the small countries ) working out with the ScaleRank vs Zoom attributes.
ScaleRank is a hint of how much "importance" the shape have. Importance could be many things like, size, population, etc.
So for example, if you want progressively display their names you need to create some rules like these:
#countries {
text-size: 0; //Text size is 0 for every country by default
//ScaleRank = 1 means it is an important country, so if the zoom level <=2 ( which means, at very far level ) starting showing it with size 12
[ScaleRank<=1][zoom<=2] { text-size: 12; }
//ScaleRank = 2 means it is quite important, so start showing at zoom 3 and size 10
[ScaleRank<=2][zoom<=3] {text-size: 10: }
}
I hope you get the strategy, you will end up doing it for many different ScaleRank and Zoom Levels.
Regarding how MapBox street works
It have been a while since mapbox lunched they vector tiles technology. With this technology the source of information ( countries, states, provinces, highway, etc ) are being consolidated in a intermediary database maintained by MapBox. You can easily customize your maps using this database because it came with this kind of features by default. If you are interested on that, you can follow the Tilemill2 development