The Anderson-Darling goodness-of-fit statistic (AD-Value) measures the area between the fitted line (which is based on an exponential distribution) and the empirical distribution function (which is based on the data points). The Anderson-Darling statistic is a squared distance that is weighted more heavily in the tails of the distribution.
Minitab uses the Anderson-Darling statistic to calculate the p-value. The p-value is a probability that measures the evidence against the null hypothesis. Smaller p-values provide stronger evidence against the null hypothesis. Larger values for the Anderson-Darling statistic indicate that the data do not follow an exponential distribution.