Suppose I really like Pete Buttigieg. He's my guy.
I want to contribute $10,000 to his campaign. Oops, that's illegal because it's above the maximum limit; the First Amendment only guarantees freedom of speech, not freedom of money, and money isn't speech. Citizens United agrees with all of that (both the majority and the dissent). Limits on campaign contributions are constitutional, including a maximum limit of $0 for corporations.
So instead of contributing $10,000 to his campaign, I'll just constantly tell everybody I know that Pete Buttigieg is awesome. This is perfectly legal. Even if Congress passed a law against it, the law would be struck down because it would violate the Constitution. I have a First Amendment right to tell people how awesome Pete is. Citizens United agrees with this as well (both the majority and the dissent).
Now suppose I want to spend $10,000 printing up some flyers about how awesome Pete Buttigieg is so I can distribute them door-to-door. Seems like a good idea except, uh oh, there's a law saying that nobody is allowed to spend more than $5,000 on flyers about politicians. I can spend as much as I want on flyers about fantasy football, but not about politicians. Is that law permissible under the Constitution, or does it violate my First Amendment right to free speech? This is the crux of Citizens United.* Citizens United says that setting a limit on how much people can spend engaging in speech is a restriction on speech. And since the restriction is based on the content of the speech (fantasy football speech is okay, but political speech is not), the restriction is not permissible. Even if I wanted to spend the $10,000 on flyers from my corporate account rather than my personal account, Congress still shouldn't be able to stop me. Not because corporations are people, but because "Congress shall make no law" doesn't distinguish between corporations and people. The majority in Citizens United sides with me on this point; the dissent does not.
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*The law in my example above is fictional. The law in Citizens United wasn't about flyers; it was about broadcasts, and contained many pages of other details. But the basic constitutional issues are the same regardless of those details.