TLDR: Where the GPU's traditional geometry API mostly worked 1 independent vertex at a time (with a few unsuccessful extensions), Mesh Shaders are a more free-form, compute-shader-like approach where you can load as much arbitrarily-structured data as you can fit in a fixed SRAM then compute however you like and output triangles directly to the rasterizer. Pretty much exactly like the VU1, but with Shader Model 6 syntax.
TLDR: Where the GPU's traditional geometry API mostly worked 1 independent vertex at a time (with a few unsuccessful extensions), Mesh Shaders are a more free-form, compute-shader-like approach where you can load as much arbitrarily-structured data as you can fit in a fixed SRAM then compute however you like and output triangles directly to the rasterizer. Pretty much exactly like the VU1, but with Shader Model 6 syntax.