Here's my attempt to distill the issue of corporate personhood. The best book on the subject that I've seen is
The Corporation : The Pathological Pursuit of Profit and Power, by Joel Bakan.
Corporations are disproportionately powerful, because they concentrate huge amounts of money and therefore political influence. Is this good or bad? On the one hand, "what's good for GE is good for America." On the other hand, corporations by law must pursue no objective over maximizing shareholder value. This means externalizing costs whenever possible. For example, a corporation sells 100,000,000 widgets whose use will kill 100 people. That corporation may not legally settle for a smaller profit in order to make its widgets safer, and thereby save the 100 people. (The "what's good for GE" proponents may counter that the public sector makes such
cold-hearted calculations all the time, and that the economic activity of the widgets is part of the "rising tide that lifts all boats," making the whole society wealthier and safer.)
What does this have to do with "corporate personhood?" Corporations concentrate huge amounts of capital, but they defuse almost all responsibility. When a corporation makes a million dollars, that money is shared by all its owners. When a corporation kills a person, that guilt is claimed by no one.
Though a corporation is fundamentally no more than a financial partnership, the existence of the corporate entity insulates its officers from morality and from the law. A bad action could spring from the combined actions of many members of a corporation, though none of the individual actions is bad. Even when a single member performs a bad action on behalf of the corporation, it's difficult to detect and prosecute. This insulation itself encourages bad behavior, because members' responsibility to the corporation is more pressing than their responsibility to the society and its laws.
Basing the rights of the corporation on the rights of the person gives corporations additional power. The law gives individuals various rights to protect them from the power of the state (e.g. the Bill of Rights). Sharing these rights with corporations gives them even more power. Treating people and corporations the same allows using a rhetoric of "freedom" to weaken and eliminate regulation, when regulation is the best tool for making corporations consider important externalities, and the only way to correct market failures.