"1 standard deviation above the average score" means that the observation value minus the population mean just equals to 1 standard deviation.The average score is the population mean.
Assume a observation selected from a population follows normal distribution (u,s^2) is x,we can calculate its Z-score = (x-u)/s = n,and we say that the observation value is n standard deviation above (positive n) or below (negative n) the average score.