In short, Clinton's job approval ratings in 1998 basically remained high in the midst of impeachment because the public perceived that Clinton had been highly successful in doing the job for which he had been elected. A primary driving force in the high job approval ratings was the positive perception of the way in which Clinton was handling the economy. At the same time, the public downgraded its evaluation of Clinton on moral conduct-related measures. Americans were well aware of Clinton's Lewinsky-related behavior, including lying under oath, but apparently came to the conclusion that this behavior was not directly related to his ability to perform his duties as president.
The fact that the public came to this conclusion is an important focus for continuing discussion and analysis. There are a number of theories that purport to show why Clinton's moral failings did not have more of an effect on his job approval rating in the eyes of the public. Most likely the 1998 impeachment crisis was unique to this particular president, this particular year, and to the particular cast of characters involved in the high drama on both sides of the issue. At some future point, a president accused of the same failings may well find public opinion strongly against him.
The public's ability to divorce this particular president's morality and ethical behavior from assessments of his ability or right to continue to govern has created in some observers consternation, puzzlement and a sense of "moral outrage." But this outrage should not be directed at the pollster's measures. The basic tools used by pollsters to measure the president, including the job approval measure, performed their function well during the crisis. The classic job approval rating, despite criticism, remains a highly important measure of the public's confidence in its president and allows the public to summarize its opinions with a measure that approximates what its vote would be if elections were held more frequently than every four years. In this situation, the verdict from the public was clear. Clinton's high job approval numbers, maintained in spite of all that he did relating to Monica Lewinsky, translated into a strong sentiment that he should remain in office.