If the correlation of the asset’s returns with the portfolio’s returns is 1, the standard deviation of the portfolio with the asset included will be a linear combination (i.e. a weighted average) of the standard deviations of the portfolio’s and the assets’s return. Since the two standard deviations are equal, if the two returns were perfectly correlated, adding the asset would not change the portfolio standard deviation (in other words, the average of two things that are the same is the same as the two indivual things).
Think of the case of perfect correlation as the “worst case” in terms of risk reduction. In contrast iff correlation is less than perfect (i.e. this case), the “new” standard deviation will be LESS than the linear combination of the standard deviations of returns. Since the linear combination of the portfolio and new assets standard deviations is the same as the portfolio’s standard deviation by itself, the “new” standard deviation must be lower than the portfolio’s standard deviation.