Recall that the probability of obtaining a sample mean as large as 580 (n = 10), when the population mean is 555 is .2843. This means that 28% of the time that we take a sample of 10 people from a population with a mean GRE score of 555, the average GRE score for the sample would be 580 or above.
The most important issue to understand here is that our sample mean of 580 is not surprising given an assumption that the population mean is 555. Since the sample mean is not surprising, we cannot claim that the program raises scores.
To have confidence that the program raised GRE scores, we would have to obtain a sample mean that was very improbable for a population mean of 555.
For example, suppose we randomly selected a sample of 10 people from another GRE preparation program and found a mean of 655 on the GRE quantitative section. Our z-score would be 2.27, p = .0116, telling us that a mean this large or larger would occur only about 1% of the time if the population mean for the program was actually 555. Since a sample mean this large would be unlikely, we would conclude that it is likely that the program does raise scores. Statistically speaking, we would reject the null hypothesis that the population mean for the program is 555.