The allegiance model asserts that part of what holds a democratic society together is the allegiance that its citizens have to their country and to its form of government. Without that, the democracy is more likely to fall apart.
In a democracy, everybody gets to speak and everybody gets to vote. This means that even people who hate democracy can participate. Such people could, if they win, destroy democracy.
Therefore, people need to have some allegiance to democracy if democracy is to succeed. They have to believe in the system even if they think that certain actors or certain policies are terrible.
To put it in today's terms, the allegiance model says that we need the Tea Party to think that democracy and our democratic system are great and that it is only the President and specific members of Congress that are horrible. As long as they hold this kind of attitude (rather than saying our whole democracy is bad) we can continue as a stable democratic system.