What did Montesquieu have to do with the constitution?

Montesquieu's relation with the U.S. Constitution has to do mainly with his idealization of "checks and balances".

A bit of background on Montesquieu's theory about government so it's easier to understand why he would prefer checks and balances:
Montesquieu wondered how different environments, histories, and religions had created such a variety of governmental institutions. He proposed a threefold classification of states. A republic was governed by many, either an aristocracy or the people. A monarchy was governed by a single authority who ruled in accordance with the law. Despotism, Montesquieu's most negative example, allowed a single ruler to govern without law and checks from other powers. A republic was analogous to virtue, monarchy to honor, and despotism to fear.

Montesquieu admired the British system with its separate and balanced powers: executive, legislative, and judicial, which prevents the government from a dangerous shift towards despotism.

Similarly, the U.S. constitution followed the concept of "checks and balances" with three different branches, executive, judicial, and legislative, checking and balancing each other. Montesquieu was key in establishing the fundamental concept on which the U.S. government was established.