Starr was winding up his investigation when the Monica scandal hit. Clinton had lied under oath about his relationship with Lewinsky in sworn testimony in the Paula Jones sexual-harassment case, and suggested to others that they lie. The very purpose of the independent-counsel statute was to keep presidents from committing such crimes against the truth. Starr wouldn't have been doing his job if he ignored Clinton's conduct in the case.
Did Clinton really lie? Of course. In the "60 Minutes" interview, he cited Judge Susan Webber Wright, who presided over the Jones case. He didn't mention that she ruled that there is "simply no escaping the fact" that Clinton gave "intentionally false" answers to questions from the Jones lawyers in a way "designed to obstruct the judicial process." Clinton's lawyer, Bob Bennett, warned him prior to the Jones deposition: "The only thing you have to worry about is if you lie in there. The crazies will come after you. They'll try to impeach you if you lie."
It wasn't just "the crazies." Nearly everyone agreed on the need to undertake an impeachment inquiry. Thirty-one House Democrats voted for the Republican inquiry plan, and the rest voted for an alternative Democratic plan. There was consensus in the country for punishing Clinton. In December 1998, Clinton himself asked to be censured. A proposed Democratic censure resolution stated he had "egregiously failed" his constitutional oath, "violated the trust of the American people" and "dishonored the office."
People wanted Clinton rebuked, but not removed from office -- and so impeachment served as a kind of monster censure resolution, expressing the country's disgust at the president's conduct and his lies. If that's something Clinton considers a "badge of honor," well then, he has a very idiosyncratic notion of honor.