Translation:The mother looks out of the window and thinks that it is going to rain later today.
Why it is "af" here and not "ad"? E.g. here: http://sproget.dk/raad-og-regler/typiske-problemer/ad-eller-af/ad-eller-af-uddybning we can see "Han kigger ud ad vinduet."
think of "ad" as the same as "via" in english. It describes the route. "af" is the same as "of". It describes where from.
There are several ways you can use a window to get a view: You could look out "through the window" or you could look "out (of) the window". Examples: "Jeg ser ud AD vinduet at postbudet er udenfor." and "Jeg sidder i vinduet og kigger ud AF vinduet".