Recently, I've read that the President has the power to pardon himself and anyone else for federal crimes. However, if he broke the law in any one of the 50 states and is indicted, he's unable to pardon himself for those crimes. What was the logic for this particular rule?

I don't know that this particular rule was consciously designed by the framers, but it is a natural consequence of the way the United States was set up to be a voluntary coalition of sovereign states.

When the United States was formed, the Constitution was written as a way for the individual sovereign states to band together and cede some of their inherent sovereign powers to a federal government. The Constitution outlined the structure and the powers of the federal government, and the federal government was to be a government of limited powers.

As Justice Kennedy observed in his concurrence in U.S. Term Limits v. Thornton, 514 U.S. 779 (1995):

Federalism was our Nation's own discovery. The Framers split the atom of sovereignty. It was the genius of their idea that our citizens would have two political capacities, one state and one federal, each protected from incursion by the other. The resulting Constitution created a legal system unprecedented in form and design, establishing two orders of government, each with its own direct relationship, its own privity, its own set of mutual rights and obligations to the people who sustain it and are governed by it.

By ratifying the Constitution, the legal theory is that the states were voluntarily agreeing to join the Union but not giving up their sovereignty. Federalist No. 39:

That it will be a federal and not a national act, as these terms are understood by the objectors; the act of the people, as forming so many independent States, not as forming one aggregate nation, is obvious from this single consideration, that it is to result neither from the decision of a MAJORITY of the people of the Union, nor from that of a MAJORITY of the States. It must result from the UNANIMOUS assent of the several States that are parties to it, differing no otherwise from their ordinary assent than in its being expressed, not by the legislative authority, but by that of the people themselves. Were the people regarded in this transaction as forming one nation, the will of the majority of the whole people of the United States would bind the minority, in the same manner as the majority in each State must bind the minority; and the will of the majority must be determined either by a comparison of the individual votes, or by considering the will of the majority of the States as evidence of the will of a majority of the people of the United States. Neither of these rules have been adopted. Each State, in ratifying the Constitution, is considered as a sovereign body, independent of all others, and only to be bound by its own voluntary act. In this relation, then, the new Constitution will, if established, be a FEDERAL, and not a NATIONAL constitution.

Any power not delegated to the federal government was reserved by the states, which retained their sovereignty, as the Tenth Amendment made clear:

The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.

So that's the background of why states are sovereign. My favorite source for reading about the origins of different constitutional principles is Akhil Amar's America's Constitution: A Biography, and I probably subconsciously drew a lot on that in writing my answer.

With that history in mind, let's look at the consequences of that framework. Because the Constitution reserves sovereignty for the states, the states could still pass their own laws (indeed, the Constitution implicitly mentions the laws of the states by giving federal courts authority to hear cases arising under federal law or cases between citizens of different states). And naturally, the power to make and amend a state's own laws includes the power to prescribe criminal punishment.

So the federal government has very little power to intervene in state criminal cases. The only area of the law where the Constitution has given the federal government to control the states, on matters of criminal law, is the Fourteenth Amendment's guarantee that the states have to provide due process of law and provide equal protection under the law to all its citizens.

What this means in practice is that states can define crimes and court procedure as they see fit, as long as they don't violate the constitutional rights of the accused. The state courts must respect the federal constitutional rights of the accused, because the federal Supreme Court has the power to review state cases and reverse state convictions for reasons of federal law (usually the constitutional rights). But the Supreme Court is not able to correct what it perceives to be errors of state law. The state's highest court is the last word on state law, even if its decisions on federal law can be reviewed by the Supreme Court.

After conviction, state prisoners can seek writs of habeas corpus from the federal courts, but they're only allowed to argue their federal rights. If they're wrongfully imprisoned in violation of state law, the federal courts can provide no relief.

The whole system leaves a lot of independence to the states to define crimes and their punishments, and the federal government has no power to intervene unless a federal law (usually constitutional) has been broken. The president is a federal officer, so the president has no power that the federal government does not have. As a result, the president can't pardon state crimes.