I invaded and destroyed fleets in 4 hyperspace anomalies. Then I declared war and cleared out the other 2. Nothing ever happened. I couldn't interact with the spheres. The Herald ran away before I could attack him.

What is supposed to happen? How do I access it?

The feature doesn't appear to be fully implemented yet. The last patch notes show that they are still adding functionality to the Arda ships. Eventually there will probably be a victory condition based on solving the mystery of the Arda and gaining control of the warp lines. For now, the main purpose of conquering the nodes is to keep Harpy fleets from coming out. Note that you can build an outpost on the nodes.