That the king is not to be trusted without being looked after, or in other words, that a thirst for absolute power is the natural disease of monarchy. (1.17)
For Paine, it's impossible to have a king as your leader and still live in a free country. In a free country, decisions should be made by leaders who've been elected by the people. Not Native Americans or black slaves, though. Paine is only talking about freedom for white men.
Did it [monarchy] ensure a race of good and wise men it would have the seal of divine authority, but as it opens a door to the foolish, the wicked, and the improper, it hath in it the nature of oppression. (2.16)
One of the reasons monarchy ruins freedom is because it gives power to a bunch of people who totally don't deserve it. In Paine's mind, it's hard to separate freedom from justice. If you're living in a society that isn't just, then it isn't really free either.
For it is the republican and not the monarchical part of the constitution of England which Englishmen glory in, viz. the liberty of choosing an house of commons from out of their own body—and it is easy to see that when republican virtue fails, slavery ensues. (2.23)
The only thing good about the English political system, according to Paine, is the fact that some of their representatives get elected. But that's cold comfort when these people still have a king and an aristocracy ruling over them.
Hither have they fled, not from the tender embraces of the mother, but from the cruelty of the monster; and it is so far true of England, that the same tyranny which drove the first immigrants from home, pursues their descendants still. (3.11)
The first people who emigrated to America are those who have sought shelter from their oppressive homelands. And that's exactly why America needs to make good on its promise and become a place of total freedom. Unfortunately, the powers-that-be in England still tend to clamp down on freedom from halfway across the world.
But the most powerful of all arguments, is, that nothing but independence, i.e. a continental form of government, can keep the peace of the continent and preserve it inviolate from civil wars. (3.38)
Freedom isn't just necessary to protect the rights of individuals. It's necessary in order to keep the people of a country satisfied with their political system. If people don't feel free, they'll eventually fall into civil war. And Paine would much rather see the Americans fight the British than see them fighting one another.
Securing freedom and property to all men, and above all things, the free exercise of religion, according to the dictates of conscience. (3.47)
For Paine, the two central freedoms of an independent America need to be the freedom to own property and the freedom to practice any religion a person wants to. When he says freedom of property here, Paine is specifically criticizing the brutally high taxes that the British made the Americans pay.
[And] until we consent, that the seat of government, in America, be legally and authoritatively occupied, we shall be in danger of having it filled by some fortunate ruffian […] and then, where will be our freedom? (4.21)
Until America is able to elect its own government, it'll never be a free country. And in the meantime, the country will just keep living with rulers who've been sent from Britain (a country that couldn't care less about the welfare of Americans).