What do we mean by the assurance, "It's not personal"? Why is that supposed to mollify us?

Great question! It might mean different things in different contexts! When a firefighter tells you this after rescuing you, she is probably trying to prevent you from thinking she is the new love in your life. "It's all part of the job" sort of thing. In the context of philosophy, the expression probably comes up when one philosopher is criticizing another. Aristotle says something like he has loyalty to Plato (his teacher for 20 years) but he loves truth more. He might have said: "Plato. My not accepting your theory of ideas and the soul is not personal." I suppose the expression conveys (on occasion) that mutual affection, even close friendship, is not a guarantee of agreement or loyalty to the views and arguments at issue. In that sense, while the expression may not "mollify" it might be intended to convey the message that disagreement does not mean personal disrespect or (even) lack of love for the persons involved.

Still, I am drawn to the (at least general) idea that philosophers should take their responsibilities personally. If I am a failure in class or seminar room and I realize I did not respect members of the class, I think I should take that personally insofar as it is a failure of myself as a person and in the practice I have devoted my life to. Perhaps one should "not be personal" in the sense that one should be "objective," but insofar as "being personal" means taking matters seriously as a person with commitments and practices, then I think it might be good if, in the course of a dialogue, someone said "it's personal" if they mean that they --as persons-- are invested in the dialogue. They are all in.

Not intending to mollify, but trying instead to stimulate further reflection.