In the traditional rendering pipeline you unpack triangles on the CPU, and then send them to the GPU, which then maps them to screenspace and then shades in the pixels to the screen buffer.
Because of modern demands we're wanting to render insane amounts of triangles.
Mesh shaders alleviate some of the bottlenecks by being a step in the pipeline on the GPU before the other steps that can generate triangles. So for example you could upload a smaller amount of triangles to the GPU, and have the mesh shaders refine that geometry by procedurally fill in the details. You could even send it not even triangles but for example the locations where you want grass, and the direction of the wind or whatever and have the mesh shaders generate all of the grass geometry.
Excuse the laymanny language, I'm just a casual observer and have imprecise memory. The tech is cool though, I wonder why it took so long for such a feature to arise, is it because there's more general purpose compute on GPU's now?