From teanglann lig means 1) let, allow, permit, 2) let go, allow to proceed, 3) let out, release, all of those imply a permissive action versus a passive action. Some form of the verb fág would be used for leaving the window open. Fágann sé an fhiunneog ar oscailt.
From reading the comments on several sentences in this lesson I gather that the Irish sentence means "He leaves the window open", in other words he allows it to remain open.
Maybe this can be expressed as "He lets the window open" in Hiberno-English, but in Standard English that means something completely different, namely that he allows it to open. Why have a principal translation that might work in one dialect of English but is categorically wrong in Standard English?