Note that Falwell sued Flynt for the tort of intentional infliction of emotional distress. In short, that means that Falwell was a typical conservative whiner. He could dish it out to gays, feminists, the ACLU, and the People for the American way, but he certainly couldn't take it.
Anyway, here's the holding (a/k/a "The Bottom Line"):
In order to protect the free flow of ideas and opinions on matters of public interest and concern, the First and Fourteenth Amendments prohibit public figures and public officials from recovering damages for the tort of intentional infliction of emotional distress by reason of the publication of a caricature such as the ad parody at issue without showing in addition that the publication contains a false statement of fact which was made with "actual malice," i. e., with knowledge that the statement was false or with reckless disregard as to whether or not it was true. The State's interest in protecting public figures from emotional distress is not sufficient to deny First Amendment protection to speech that is patently offensive and is intended to inflict emotional injury when that speech could not reasonably have been interpreted as stating actual facts about the public figure involved. Here, respondent is clearly a "public figure" for First Amendment purposes, and the lower courts' finding that the ad parody was not reasonably believable must be accepted. "Outrageousness" in the area of political and social discourse has an inherent subjectiveness about it which would allow a jury to impose liability on the basis of the jurors' tastes or views, or perhaps on the basis of their dislike of a particular expression, and cannot, consistently with the First Amendment, form a basis for the award of damages for conduct such as that involved here.
And here's some background on the case for those who prefer not to slog through Supreme Court prose.