I think the excerpt needs to be taken in the context of the previous paragraph and book 6 in general. Book 6 deals with the fundamental notions and foundations of democracy. In section II, there are two primary principles of democracy, that in one instance, democracy and its principles should be observed (like equality, majority rule, etc.). The second is that there be true autonomy for a citizen.

In the section you quote from (AP, 6, II,P2), the issue in question is the position of the electorate and appointments to office. This is in essence term limitations, surety of integrity in office, etc. From this point it gets to where your question is;

"Of all magistracies, a council is the most democratic when there is not the means of paying all the citizens, but when they are paid even this is robbed of its power; for the people then draw all cases to themselves, as I said in the previous discussion."

In the early fifth century, Solon becomes the head Archon (of the three) in Athens (following Drakon some time before). He introduces a series of reforms in the wake of the Damasias affair (of which there is very little information about, but this helps to date the timeframe in which the Solonic reforms are established, which is around the 580's). Among the various reforms Solon imposes, such as the end of the seisaktheia (shaking off of burdens) imposed on destitute citizen farmers (Gorgoi), Solon establishes a series of class structures. This class structure (composed of valuation of property measured in bushels of wheat) forms the foundation of a class structure that Aristotle is so influenced by in this quote. Solon "revolutionizes" the way in which the Athenian city state is organized, by creating a board of 9 archons (an archon, a polemarch, an archon basileus, and 6 theomthetai). In the same fell swoop of reform, Solon reforms the Council of Athens, the Boule of 400. Things progress, Kleithenes comes along, reforms the social and political system again, creates the 30 Tryttes and Demes (local units) and reorganizes them into phyles. It is from these groups that the new Boule of 500 is composed.

The Boule (the council) is of vital importance in all of this. It is this assembly which constitute what we would call a senate, a democratic convention composed as a representation of all the people of Athens. Payment wise, around the 460's, Herekles (new Archon) introduces a stunning law? pay for public service. Aristocrats do not like this at all. With payment for public service, this makes it possible for everyone to take part in the democratic process. Before, the Boule could only be attended by those of the higher ranks (the ones established by Solon and elaborated on by Kleithenes). But now the lower ranks of the Zeugetai and Thetes (lower classes) could join in and have a say in government by lot.

However? the aristocracy does not want pay for public service. This is in essence robbing the aristocracy of their power base. Now Aristotle puts in his two cents on the issue. What the line (from as far as I can interpret it) is that because the lower classes have access to the Bouleuterion (democratic process), the "this" (the boule) is robbed of its power. It is robbed of its power because in true aristocratic form, poor people don't think on the level of the enlightened aristocrat. They think about the harvest and not about the macro-level affairs of the city state. By withholding pay for public service, the right people can hold office.