What to do with it after it has been ground out?
Grinding reduces the stump to small wood chips suitable for many purposes within your garden.

Mulch
As a mulch layer around trees and plants it will reduce evaporation from the soil, keep the soil a little warmer (important for root development) and suppress weed development. Any weeds that do germinate on the mulch will be easily tweaked out of the loose surface.

Soil Improvement
Because of the open texture and high surface area of the material it will hold water in the soil which will be available to plants. Depending on the species of the original tree the material will decompose slowly in the soil making natural resources available to plants.

Compost
Placed in a compost bin or loose heap the material will decompose to a high quality humus for use in the garden.