What political issue did the three-fifth's compromise in the Constitution attempt to address?

The southern colonies feared that in any new government designed as a representative democracy, they would lose power because the populations of their states were comprised in large part of slaves, who could not vote. Northern states, by contrast, had very few slaves, and their populations were mostly free, meaning that if representation in congress was proportional and based solely on the number of free men in each state, then the voting power of the southern states would be massively diluted. The southern states did not want to lose their political influence, and refused to sign onto a constitution that would relegate them to being lesser powers with many fewer representatives in the federal legislature.

So the southern states proposed that for purposes of allocating representatives to the House of Representatives, slaves living in their states should be counted. Yet delegates to the Constitutional Convention from the northern states rejected this plan, because it would give the slaveowners in those southern states vastly more representation than any free men in the north, and would tilt the balance of power in the Congress toward southern states and their economic agenda, which were at odds with the agendas of the northern states.

In order to satisfy the southern states that their interests would be represented in the new government, and in order to quell the fears of less populous northern farming states, like New Jersey, delegates to the Constitutional Convention came up with the 3/5th's rule. This rule stated that although slaves would not get the right to vote, be citizens, or have another individual rights, for purposes of allocating representatives in the government, each slaves would be counted as 3/5th of a person. This would give the southern states more representatives than if slaves were not counted at all, but not so many representatives as if each slave were counted as a full person. 

The irony of this decision is that this rule benefitted white plantation owners with thousands of slaves at the expense of both slaves and poor white farmers. The voting power that accrued to states like Virginia and South Carolina as a result of the 3/5th's law was incredibly anti-democratic, and went to perpetuating the institution of slavery. Had the 3/5th's law not been established, slavery might have been abolished much sooner, through votes in a less deadlocked congress, and the country might have avoided the Civil War.