One summary of the details is here (I don't know whatever inside story there may be), but the bottom line is that he had to resign from The Washington Post because of negative comments he made about conservative figures on a supposedly private email list. Weigel's job for the Post was to cover the conservative movement and it seems the Post expected him to maintain a more objective stance, including in his private emails. Matt Yglesias has more extensive coverage of the episode and here is Ross Douthat. Here is Weigel's account and apology, which includes the postings which got him into trouble. And here is a detailed Politico article.
It is likely I prefer Weigel over his replacement, and if you're wondering I don't know Weigel well, even though he lives nearby.
At a more general level this is a tax on journalists, who now have a greater fear of being fired for past actions. It's also a tax on the moody, the volatile, the web-savvy, the non-mainstream, and a subsidy to in-control smooth talkers and careful writers.
The Washington Post wrote:
“But we’re living in an era when maybe we need to add a level” of inquiry, he [a WP web site managing editor] said. “It may be in our interests to ask potential reporters: ‘In private… have you expressed any opinions that would make it difficult for you to do your job.”
I'm not sure what kind of answers they expect to that question nor what they understand by the word "private" in that context.
Conceptually, the core problem is that the distinction between the private and public spheres is breaking down, but at different rates for individuals and mainstream institutions. The practical question is what an equilibrium would look like for the WP, given that the paper courts advertisers, relies on political contacts, and wishes to avoid becoming a target for right-wing (and left-wing) media. It's easy to imagine the targets of Weigel's criticisms citing them repeatedly against the Washington Post and questioning the Post's objectivity. "Oh, that was written by the guy who said that…"
One possible outcome is that the current public code of behavior becomes applied to writers' private lives and I suppose that is what we are seeing and it is also what a lot of "common knowledge" models would predict. That is, most of us know that many writers say such things in private, but that's tolerated as long as it doesn't become common knowledge about any particular writer.
Common knowledge mechanisms often lead to inefficient (and unfair) outcomes, in part by non-convexying returns with regard to the actions of the individual. Maybe we would like taxes to be linked to individual type, but common knowledge mechanisms tend to link the actual "tax" to how social forces process information about an individual. A polemicist who is secretly taped encounters a greatly different outcome.than a polemicist who is not taped.
One option is for public institutions to adopt a "statute of limitations" for private remarks and with a short time window. That would not help in this case, since Weigel's relevant postings do not predate his Post employment; still it might be a good reform. Another option is for public institutions to adopt different norms for their web writers. There are already different norms to some extent (web writings receive less editing, for instance), but it is hard to spread the different norm for the writing to become a different norm for the writer. Web traffic is already massive for newspapers, and most readers probably do not distinguish between different kinds of paper employees, such as web vs. non-web. Anyway, it's a fuzzy line if a writer has both web and non-web output.
A more radical change would move away from the manufactured image of the objective newspaper, but this is especially difficult for the Post, given that it relies on both conservative and liberal sources for its key political coverage.
Overall, we need more incentive-compatible, generalizable organizational reforms which will allow mainstream institutions to have more flexible relationships, and indeed sometimes more distanced relationships, with their writers. Yet reputational forces are often quite blunt, and grossly calculated, and mainstream institutions are not very far along on making such reforms work.