What is the basis for the expiation of fasting forty days for those who watch dubbed films, and what is the sin that necessitates it, and how can a Muslim atone for watching many of these series?

The obligatory is sincere repentance, accompanied by remorse and cessation of the sin, and a firm resolve not to return to it. This should be coupled with abundant seeking of forgiveness (istighfar) and performing righteous deeds, for "indeed, good deeds obliterate evil deeds." This will atone for the sins of unlawful looking and listening. You are not required to offer a specific expiation for watching films unless there is a textual religious proof for it. Fasting, in general, is among the good deeds that obliterate evil deeds.