There are a couple things at work here. One is the structure of the molecule cellulose.

Cellulose is a very long molecule and is almost woven together in the wall (see this image for a cross section:

plant cell wall

This gives it a lot of what's called tensile strength: it is strongest when it is being pulled.

So for example, a string has tensile strength when you are pulling on either end, or a balloon has tensile strength the more water you put in it and the more it stretches out. The most important aspect of the cell wall remaining rigid is actually exactly like that. Plant cells get stuffed full of water to keep the cellulose in the walls tight. This makes them rigid, and is also why plants that haven't been watered will lose their stiffness and droop.