What happens is this: Rivers wash away dirt and sediment from the lands they pass over. Much of this ends up at the river's delta, where the waters spread out and slow down before joining the sea. Over time, sediment can pile so heavily it can actually cause the Earth's crust to start sagging underneath, maybe an inch or so downward every decade. What's more, as this sediment landscape dries out it compresses, sinking even more. Those muddy waters of the Mississippi have left behind such a weight of mud that New Orleans is currently below sea level. If river deltas sink, what keeps them from going under? After all, the sea is right there, waiting to flood any land that sinks too low.
The answer is all that sediment. Although the Earth's crust sags under its weight, the river always has new sediment to deposit--especially when the delta is allowed to flood periodically. This rising sediment keeps pace with the sinking crust, and the level of the land stays pretty constant.
Sinking deltas become a problem if the river's waters are dammed or channeled, rushed to the sea without a chance to spread new sediment over the wide, delta valley. This lack of new sediment is what causes New Orleans to be below sea level.