Here's the "atrocity" page, in which Horsley reprints from a pro-abortion site a list of abortion providers including annotations as to which have been killed or wounded. His argument is that when he did that, he (well, the ACLA) got sued. Further down that page, Horsley uses the strikethrough mechanism to list women who have died during abortions. (Abortions have genuine risks. Childbirth does too; it used to be conventional wisdom that maternal risk from abortion was much lower than maternal risks from carrying a baby to term.)
Here's the page of Horsley's containing tables from before and after Dr Slepian's assassination. The day before, according to Horsley, his only intent was that cited above; the day after, he added Dr Slepian's name, with a strikethrough. Slepian's name had not been there before, leading Horsley to protest very strongly that his site could not have been a threat against Slepian. (The lawsuit was filed by other physicians who felt it was a threat to them; Horsley is silent on this.)
Planned Parenthood claims the site "celebrate[s] violence against abortion providers".
Horsley is having trouble finding ISPs willing to host his site. The notion of ISP censorship is an interesting one in its own right.
Here's a good site with background information about the case. The actual lawsuit was brought in 1995 by Planned Parenthood (and some abortion providers) against American Coalition of Life Activists (ACLA) et al. Horsley was not party to that suit; his Nuremberg site was in fact not created until 1997. The original lawsuit was over threatening "Wanted" posters depicting abortion providers; Horsley's site (but not Horsley himself) was added later. In retrospect it seems reasonable to think that, if it were not for the context created by the "Wanted" posters, there would have been no issue with the Nuremberg Files web pages.
The central question in the case is whether the statements amounted to a "true threat" that met the standard for being beyond the bounds of free-speech protection.
The judge gave the jury instructions to take into account the prevailing climate of violence against abortion providers; the jury was also considering not an ordinary civil claim but one brought under the Freedom of Access to Clinic Entrances act (FACE), which allows lawsuits against anyone who "intimidates" anyone providing an abortion. (The first-amendment issue applies just as much with the FACE law as without.) The jury returned a verdict against the ACLA for $100 million, and the judge granted a permanent injunction against the Nuremberg Files site (Horsley's). Here is that injunction.
Under current free-speech standards, you are allowed to threaten people. You are allowed to incite others to violence. You are not allowed to incite anyone to imminent violence, and you are not allowed to make threats that you personally intend to carry out.
At first glance, the jury's decision was inconsistent with these standards. When the case was heard on appeal by the Ninth Circuit Court, by a three-judge panel, the jury decision was overturned.
Here is the initial appellate-court decision, written by Judge Kozinski. He bases his decision largely on the case of NAACP v Claiborne Hardware, a Supreme Court decision from 1982. The NAACP had organized a boycott of some white-owned businesses, and had posted activists to take down names of black patrons. The NAACP liaison, Charles Evers, had stated publicly that those ignoring the boycott would be "disciplined" and at one point said "If we catch any of you going in any of them racist stores, we're gonna break your damn neck." The Supreme Court found in the NAACP's favor, on the grounds that there was no evidence Evers had authorized any acts of violence, or even made any direct threats (eg to specific individuals). Kozinski argues that whatever the ACLA was doing was less threatening than what Evers was doing, and therefore dismisses the case.
Then the entire Ninth Circuit Court (not just a three-judge panel) agreed to hear the case (this is called an en banc hearing). They reversed again, upholding the original jury award.
Here is the decision of the entire Ninth Circuit Court. It's long; they took this seriously. It includes a detailed finding about the facts of the case. The decision makes a distinction between "advocating violence" (protected) and "threatening a person with violence" (not). They also argued that context is all-important, and that this is why the Claiborne case above is not a good parallel. In that case, despite Evers' threats there was relatively little violence: no one was seriously injured, and some black members of the community continued to patronize the stores. So, in that context, Evers' statements were not "true threats". The context here, where several abortion providers had already been killed and where harassment was continual, was entirely different.
Here is their discussion of that context, pertaining to the original WANTED posters rather than the website:
ACLA appealed to the Supreme Court. While there was still some hope the Supreme Court might see things differently, the Ninth Circuit had established that the speech in question met certain standards for being a true threat, and the ACLA would have had to argue that some factual interpretations were mistaken. But the Supreme Court does not generally decide cases about facts; they accept cases about significant or conflicting legal principles. As the principles were nominally settled by the Ninth Circuit, the Supreme Court did not agree to hear the case.
Finally, you might wonder why, with all the threats of violence made during the course of the civil rights movement by whites against blacks, the case NAACP v Claiborne that comes to us is an allegation of violence by blacks against blacks. I think it's safe to say that the answer has nothing to do with who made more threats, and everything to do with who could afford more lawyers.