The height is just the height of the two-by-four, so about 3 1/2 inches. When the hens step onto the frames, the mesh bows down a bit in the middle, so once the grass is about 3 inches tall, they can start nibbling on the tips.
Mostly, they're just nibbling off the tips that poke through, but depending on how vigorously they tear at it and how firmly rooted any given blade of grass is, they will occasionally uproot a blade. Most of those whole blades get eaten right away, while others end up forming a thatch. Towards the end of the post, I talk about that some. You can see a photo of a few frames that have been worked over for a few months. Still plenty of grass there. Hope this helps.