My Naive First Amendment Free Speech Question

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NPR's "Morning Edition" ran a long segment today on the Supreme Court's rehearing of the free speech case, Citizens United v. Federal Election Commission (FEC). Citizens United wanted to broadcast on a cable TV channel a documentary that was highly critical of Hillary Clinton during the Democratic presidential primary campaign last year. However, because the group had received a donation from a corporation, the documentary ran afoul of federal campaign finance laws and FEC regulations that prohibit political advertising funded by corporations. 

According to the NPR segment:

The question always is: Who does the First Amendment apply to? Do only individuals have the right of free speech? Or does this right extend to corporations and unions as well?…

"Does it apply to foreign nationals? Does it apply to the government of China or Russia or Iran in this country? Does it apply to corporations? Those are all different players who are not individuals, not voters, not citizens," says [Trevor] Potter,[former chairman of the Federal Election Commission and a longtime McCain adviser].

While the NPR segment noted that the First Amendment protects free speech, the report never quoted it. The First Amendment reads:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances. 

It seems to me that the legally relevant section reads:

Congress shall make no law … abridging the freedom of speech, or of the press…. 

So my naive constitutional question is: What part of "no law" don't the courts, NPR, and campaign finance and speech "reformers" understand? 

And actually I do think that "no law" applies to speech (and especially to political speech) by anybody or any group, including unions, corporations, foreign nationals, or Martians.

See my colleague Jacob Sullum's much more sophisticated column "Unfair, Unbalanced, But Free" parsing the issues in Citizens United v. FEC.