What was the original purpose of the 14th Amendment?

Among other things, to make sure that the government did not exclude (or perpetuate the exclusion) of people born in this country from the rights and protections of citizenship. You can't have two different groups of people who are born here: those who have the rights of citizens, and those who don't. So everyone born here is a citizen - as long as they're not outside the jurisdiction of the government (like a foreign ambassador who is already completely protected from the government by virtue of their diplomatic status), they are given the most basic protection against the government, the right to have citizenship.

That was one of the depredations of chattel slavery - it was a condition that was inherited. If the parents of a child were slaves, and therefore not citizens, then the child would themselves not be a citizen and be a slave. So that goes away - no matter what the status of your parents, everyone in this country starts off as a citizen - the core protection against government abuse of them.