The 3/5ths compromise is often misunderstood. It doesn't say that anyone is 3/5 of a person or gets 3/5 the rights. It dealt only with a state's representation in Congress. It's helpful to look at what the relevant portion of the Constitution actually says, then do a math problem:

Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons. The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at least one Representative

Other persons means slaves. This passage is describing how states will be represented in the House of Representatives.

While each state is represented equally in the Senate, regardless of population, representatives in the House are apportioned according the state's population so that bigger states are more powerful. Originally, each state got one representative for every 30,000 free residents. Since slaves counted as 3/5th of a person, states also go t one representative for every 50,000 slaves. Slave states would have wanted slaves to count towards the population the same as a free person, but northern states would have wanted slaves to not count towards the population at all.

To see why, it's easiest to look at how the different plans would have affected a hypothetical state. Pretend that there was a hypothetical slave state with 90,000 slaveholders and 150,000 slaves. How many representatives would they have under various plans?

Southern plan (1 rep per 30k people, slave or free): 240,000 total people/ 30,000 people per rep = 8 representatives Northern plan (1 rep per 30k free people, no representatives for slaves): 90,000/30,000 = 3 representatives 3/5 compromise (1 rep per 30k free and 1 rep per 50k slaves): 90,000/30,000 + 150,000/50,000 = 3+3 = 6 representatives.

The Northern states benefited from the compromise because if slaves had counted as a whole person while apportioning representatives, slave states would have gotten more representation.

The South probably got the better part of the deal though. While slaves were "represented" in Congress, they did not have any voting rights (or many other rights at all). Their owners not only exploited them, but got additional representation for doing so.

(If this is confusing, think of the hypothetical state from before. The 90,000 slaveowners with 150,000 slaves get as many votes as 180,000 non-slaveholders.)