Why did Robb Stark invade the south rather than hold the north?

From what I've been seeing and reading Robb Stark's plan does not make ANY sense.

What is the point of attacking the south rather than just holding the north? At the end of the day control of a land is only there if there are men to hold it. Why not just stay in Winterfell and rule the North as it's own kingdom? Perhaps build a wall or check point or two on the southern border to hold off southern invading forces.