How can I interpret “talk somebody out of” in “He talked the mutawwa out of beheading or lashing me”?
To talk someone out of something means to "persuade someone not to do something". A related phrase is to talk someone into doing something, which means to persuade them to do it.
In this context, Jubeir is trying to persuade the mutawwa not to punish the writer for wearing what they considered indecent clothing. Apparently the persuasion was difficult, for Ms. Dowd records that it "took a surprisingly long time".
And yes, talk someone out of something is a rather popular idiom, used in many contexts.