"Complaints are everywhere heard from our most considerate and virtuous citizens, equally the friends of public and private faith, and of public and personal liberty, that our governments are too unstable, that the public good is disregarded in the conflicts of rival parties, and that measures are too often decided, not according to the rules of justice and the rights of the minor party, but by the superior force of an interested and overbearing majority. However anxiously we may wish that these complaints had no foundation, the evidence, of known facts will not permit us to deny that they are in some degree true."
He makes the point that it is difficult to combine both majority rule with a reasonable protection of minority rights, and he references other writings that argued that previous democracies had failed because a too permanent majority eventually encroached the rights of the minority to such a degree that the minority became a supporter of any agent or movement that might improve their situation, a situation that could only be improved by ending the rule of the infringing majority.
Madison then made the argument (wholly novel at the time) that a large and diverse nation was better suited to majority rule than a small and homogenous nation, for a large and diverse nation would be less likely to have a permanent majority.
The ruling majority usually has the greatest respect for the rights of minority factions only when that majority is about to lose its majority status, and therefore perpetual changes in the majority is the best way of keeping alive a wide spread respect for minority rights (that is, if everyone can reasonably expect to be in the minority at some point during their life, then everyone will be more likely to respect minority rights).
Some people read #10 and seem to think that Madison is arguing that people should not engage in factions (that protest movements are bad for the nation) however he sets that aside very early in the essay:
"There are two methods of curing the mischiefs of faction: the one, by removing its causes; the other, by controlling its effects. There are again two methods of removing the causes of faction: the one, by destroying the liberty which is essential to its existence; the other, by giving to every citizen the same opinions, the same passions, and the same interests. It could never be more truly said than of the first remedy, that it was worse than the disease. Liberty is to faction what air is to fire, an aliment without which it instantly expires. But it could not be less folly to abolish liberty, which is essential to political life, because it nourishes faction, than it would be to wish the annihilation of air, which is essential to animal life, because it imparts to fire its destructive agency. The second expedient is as impracticable as the first would be unwise. As long as the reason of man continues fallible, and he is at liberty to exercise it, different opinions will be formed. As long as the connection subsists between his reason and his self-love, his opinions and his passions will have a reciprocal influence on each other; and the former will be objects to which the latter will attach themselves."
That is to say, faction is natural and can not be eliminated from a free society. The only question is how to channel it in constructive ways.
Over the last 3 centuries most Western nations have experimented with majority rule combined with various restraints. Some societies achieved reasonable compromises of majority rule plus some protections for those in the minority. The biggest failures of the model have been in those cases where some groups were able to see themselves as holding a permanent majority, and most of these failures have had to do with race. The writer of the linked article seems aware of this, although they do not explicitly mention it.
I think democracy is pretty great actually. I think we just screwed up requiring arguing from some sort of moral-free first principles.
A pure democracy is a joke. A Republic was a fine idea, but the US has done a really poor job of understanding the notion of a sovereign citizen (as opposed to whatever king .. er president is in power)