And corporations are people, of course
It's just a legal convention to help think about how corporations fit into laws not explicitly written for them. For example, who can enter legal contracts and what are the repercussions if they are broken? Well contract law was thought of around people, but if you think of a corporation similarly, then the law still makes sense. Of course, they can and do specify restrictions that only apply to corporations or opt them out.
A better summary is that restrictions on money being spent on speech are effectively restrictions on speech, itself, and the freedom of speech is not limited to individuals.
I didn't know Colbert coined "corporations are people" -- TIL.