The reason you don't understand it is that it is a liberal talking point, and therefore isn't exactly true.
Corporate personhood only allows the corporation to enter into contracts. Suppose you contracted with Bill, of Bob and Bill Roofing, to fix your roof. You gave him a $2500 deposit. The next day, Bill accidentally falls off another roof and dies. In your corporate-free world, Bob would have absolutely no obligation to perform as per the contract. After all, you had a deal with Bill, and that deal died along with him.
But because we actually have a corporate entity, in reality your deal wasn't with Bill. Bill signed the contract on behalf of the corporation. Meaning that Bob has to not only fix your roof, he even has to do it at the price Bill agreed to.