Mesh shaders in a way are a new version of what the Playstation 2 did back in the day, where you had a vector processor that would output triangle lists to be rasterized. Its flexible mesh pipelined allowed very interesting techniques, like the multi-layer meshes for fur in shadow of the collosus, or multiple types of smooth LOD interpolation seen in other games. With them, now we can emit geometry from a programming model that is more or less normal compute shaders. Each threadgroup will output a small mesh directly into the rasterizer units of the GPU, so the flexibility is quite great.
Game developers are running into the limits of the current vertex shader model, as you can see with trickery like doing mesh processing in compute shaders and writing the final indices into a big linked list through parallel compaction and similar. With the mesh shaders, all that becomes both more optimal and simpler to write. Techniques like Nanite become a fair bit faster using mesh shaders than emulating them by writing indices into memory, as it avoids that whole memory and syncronization round trip. Completely removing the disliked geometry shaders is also a great feature.
Its great to finally have a crossplatform implementation of Mesh Shaders, but i think the implementation leaves a bit to be desired given it seems to be a direct translated vulkan version of the DX12 implementation. Im not sure how its going to scale to hardware other than the 2 big desktop vendors. The biggest issue with the mesh shaders here is that the cache sizes or recomended amounts vary depending on the GPU. To take the most advantage of mesh shaders and get great speed out of them, there are some fast-path counts for triangles,vertices, and shader memory usages. Those vary by the vendor, so im not sure how are developers going to deal with that correctly other than having multiple versions of the shader by vendor.