The three-fifths compromise entered into the issue of slavery in Missouri because it meant that the House of Representatives had more members from Northern states than from the slave states. This clause in the Constitution, it could be argued, reduced Southern representation in the House. It declared that the full number of slaves in a state would not be used to determine the population of that state for the purposes of representation in the House. Instead, only 3/5 of the number of slaves would be counted in determining the state’s official population for this purpose. This meant that the slave states did not have as much representation as they would have if all the slaves had been counted. For example, both Louisiana and South Carolina had populations that were around 50% slave. What that meant is that about 20% of their populations (40% of the slaves) were not counted. That reduced their representation in the House.
Congress was the body that got to decide whether Missouri would have slaves when it became a state. The makeup of the House of Representatives was influenced by the 3/5 Compromise. Therefore, this compromise entered into the debate over slavery in Missouri by helping to determine the makeup of the House as it conducted that debate.