To perhaps better explain the "why" behind this rule of thumb, first think of what it means when the P/E ratio changes. If the P/E ratio increases, then this means the stock has become more expensive (in relative terms)--for example, an increase in the price but no change in the earnings means you are now paying more for each cent of earnings than you previously were; or, a decrease in the earnings but no change in the price means you are now paying the same for less earnings.
Keeping this in mind, consider what happens to the PE ratio when earnings increase (grow)-- if the price of the stock remains the same, then the stock has actually become relatively "cheaper", since you are now getting more earnings for the same price. All else equal, we would not expect this to happen--instead, we would expect the price of the stock to increase as well proportionate to the earnings growth. Therefore, a stock whose PE ratio is growing at a rate that is faster than its earnings are growing is becoming more expensive (the price paid per cent of earnings is increasing). Similarly, a stock whose PE ratio is growing at a slower rate than its earnings is becoming cheaper (the price paid per cent of earnings is decreasing). Finally, a stock whose P/E ratio is growing at the same rate as its earnings are growing is retaining the same relative valuation--even though the actual price of the stock may be increasing, you are paying the same amount for each cent of the underlying company's earnings.