There are differing views on the true purpose of the social contract, but the argument he is presenting is that we have natural rights, i.e. rights that precede any human construction or even really humans themselves, and the function of a social contract is to create a social framework (particularly laws, including legal rights, i.e. rights constructed by humans for governance) that allows us to make sure our natural rights are observed.

You are right about social contract frameworks normally being based on unpersoning, which brings into question how seriously the social contractors are really taking those supposedly-transcendent natural rights, but I think the more fundamental issue really is what the commenter in the OOP says, which is that natural rights are literally just totally made up and fictitious.