What does integrity mean?

It means the willingness to admit to an opinion, and an opinion is a bias. Readers will trust you more if they know you're telling them what you think, not what you think they should know. The article discusses a piece by Walter Pincus. You can read it here.

Again and again:
Objectivity being impossible but engagement a requirement, it's simplest to say that the job of the press is to dig for everything. It should be less their job to discriminate as to what they're digging for than to specialize as to who is digging for what: a political hack has neither the time nor the connections to cover movie stars. But If the press had shown as little respect for George W. Bush as it has for Britney Spears we'd all be better off. It should never be a question of demanding seriousness when the institutional rules for what defines seriousness always lag behind the reality. That's how culture works. You could say it's a matter of economics, that there's money in gossip and the press should choose money over high principle (and that's true), but it's more complex than that. Vulgar economism says the adversarialism of the market is foundational, but won't accept that society demands the market itself have an adversary, a counter-force to the economic telos.
There can be no single telos in democratic society, but greed is an astringent. It serves a purpose other than the one it defends. That's the most important detail.