In The Name of Allaah, The Most Merciful, The Bestower of Mercy.
The wisdom behind [الاستغفار – (i.e. saying: أستغفر الله three times)] after Salaah is to manifest the soul’s wrongdoing – that the slave [i.e. the performer of the prayer] did not fulfil the rights of the prayer, neither completely nor perfectly in the manner it is obligated. It has to be the case that he did fall into some deficiency, shortcoming and negligence; therefore, he seeks forgiveness in the hope that his shortcoming is relinquished and so that his Istighfaar [i.e. seeking forgiveness] can be a means of remedying the deficiency or shortcoming in his prayer.
An Excerpt from ‘Fiqhul Ad’iyah Wal Adhkaar’ page 171′ parts 3&4