The process itself is simple. You dig around the edge of the clump, loosening it up repeatedly by using your shovel as a lever. Eventually you work it out. The hazards are roots and rocks. The roots prevent you from digging and fight you while leveraging. The rocks just fight you.
When all the grass gets remove, the apple trees will get more good stuff for growing (assuming that they survived the winter).
I'm going to leave the root clumps to dry out. I'll clean them up in the autumn because they will be totally inaccessible by early summer as the plants in that area run riot.
I pruned my way through the other side of the yard, clearing out small bushes and stray trees. At this point, I am almost caught up to where I want to be in terms of routine maintennance. I still have many roots to dig out, but the understory is no longer quite so crazy.