Officers of a publicly traded company have a fiduciary responsibility to their shareholders to act in the company's best interests, placing shareholder interests ahead of their own. It would seem that they decided continuing to air programming that could potentially be viewed as violating FCC rules and risking their license was not in their shareholders best interests…
Or simply as Nexstar’s broadcast division president said in the article: “Mr. Kimmel’s comments about the death of Mr. Kirk are offensive and insensitive at a critical time in our national political discourse, and we do not believe they reflect the spectrum of opinions, views, or values of the local communities in which we are located.”