The 2d Amendment: "A well regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed."
The prefatory clause is a clause which announces the general purpose of the operative clause. The operative clause is the clause which creates the substance of a fundamental right.
So, imagine a constitutional provision such as this:
Prefatory Clause: "The ability of parents to support their children being necessary to a free society,"
Operative Clause: "the right of the people to own, use, and possess private property shall not be abridged."
How should we interpret the right of private property? As covering only the right to support children? Does it protect a single person's right to own private property? Does it cover only enough property to provide minimal support for children? Does it cover the right to own stocks? Savings accounts? Automobiles? A pension for retirement? Jewelry? Does it cover most lawful kinds of property owned by any American citizen?
Suppose the government passed a law confiscating all wealth above a certain minimum amount (say, above $25000). If a childless person challenged this law, how should the Court rule?
Does the prefatory clause simply tell us why the Constitution seeks to protect broad private property rights for all? Or does the general purpose define the full scope of the right?
