What is the basis for saying “astaghfirullah” after the obligatory prayers?

It is Sunnah to seek forgiveness (istighfar) after the obligatory prayers. This is because, after completing the prayer and giving salam, the Prophet Muhammad (peace be upon him) would seek forgiveness three times and say, “Allahumma anta’s-salam wa minka’s-salam…” (Muslim, Masajid, 135 [591]).

By making istighfar, one asks Allah for forgiveness for any shortcomings in the prayer. Therefore, it is in accordance with the Sunnah for the imam and congregation to individually say “astaghfirullah” after completing the prayer.

Source: T.C. Diyanet İşleri Başkanlığı, Din İşleri Yüksek Kurulu