The first two forty-day periods of prayer were of no avail, and it was only after an additional forty days that God said to Moses "I have forgiven according to your request." Therefore, we recite selichot during this same last period of forty days.
Though God accepts the repentance of his children all year long, the forty days between Rosh Chodesh (New Month) Elul and Yom Kippur are more suited for repentance than any other period of the year, for they are days of Divine mercy and favor.