I presume it's because getting up to your shins in mud makes it hard to draw your leg back out, whereas on stilts all your muscles and joints are above the stilt platform, allowing you to use them freely to draw the stilts out of troublesome terrain. That's just a guess though.
Yes, and the thinness and smoothness of the stilts allows them to slide easily out of muck whereas legs with boots and clothes on them get stuck rather easily!