Dr Bill, the "affirmative" in this quote identifies the obligation as one requiring affirmative conduct; that is, it is something which the person must do. A (not modified with the term affirmative) obligation can encompass an obligation to refrain from certain conduct.

Perhaps an example from the field of negligence law would help illustrate this. An ordinary person, going about his business on a public street, has no affirmative duty to come to the aid of another. If say, that person see another has fallen and injured himself, the person has no legal (disregard moral obligations here; that's a different subject) duty to help the injured person. There is no affirmative obligation toward the injured person. However, if the person decides to come to the aid of the injured person, he has an obligation to not act in a way which would exacerbate the harm. So, in the example, if the person decides to help the injured party but in his ignorance moves the injured parties' broken leg and causes a much greater injury, he could be liable for that extra injury. As a volunteer rescuer, the person has an obligation (not affirmative) to refrain from carelessness causing extra injury.

Does this make sense now?

The use of the term "affirmative" does not relate to the burden of proof in this context. Rather, an "affirmative defense" is one which is pleaded by a civil defendant to avoid liability despite the plaintiff's presentation of a prima facie claim. The burden of proof is on a civil defendant to prove an affirmative defense. As to a criminal defendant, an affirmative defense imposes an obligation to present some evidence in support of the defense, after which the prosecutor is required to disprove it beyond a reasonable doubt; the ultimate burden of proof never shifts in a criminal case.

edited to fix typo