As you say, the Declaration of Independence has these 4 parts.
The first part is the declaration of natural rights. It says that people have these rights by the fact that they are human. These are not rights that the government gives them -- they have them automatically because they are people.
Then you have the preamble. It gives the reasons why the colonists think they have the right to rebel. It says that if their government doesn't protect their rights, they have the right to rebel. The third section gives all the ways that (Jefferson claims) the king has failed to protect the colonists' rights. This ties in to the previous section and gives the specific things the king has allegedly done wrong. Finally, the fourth section comes out and says "so now, because of this stuff, we are our own country."