At first blush, the hints seem a bit incongruous. Let's take them one at a time. First, we know that the product of the three ages is thirty-six. The following sets of three numbers have a product of thirty-six:

{1,1,36}, {1,2,18}, {1,3,12}, {1,4,9}, {1,6,6}, {2,2,9}, {2,3,6}, {3,3,4}.

This of course does not allow the salesman to determine the ages. The second hint is that the sum of the ages equals the number on the house next door. Summing the ages, we have the following.

1 + 1 + 36 = 38
1 + 2 + 18 = 21
1 + 3 + 12 = 16
1 + 4 + 9 = 14
1 + 6 + 6 = 13
2 + 2 + 9 = 13
2 + 3 + 6 = 11
3 + 3 + 4 = 10

After looking at the number of the house next door, the salesman still does not know the ages. This means the ages must have been one of the two sets that sum to 13 (otherwise he would have know which set to select). Since the woman then said "The oldest one plays the piano," the only possibility is the set of ages {2,2,9}, since the set {1,6,6} would imply that there are two oldest children.