Who specifically?
Those advocating banning DDT: Chiefly, the Environmental Defense Fund and EPA Administrator William Ruckleshaus.
And what details do you have to explain why they did this? What clearly motivated them, and what may have been additional motivations?
That would seem to me redundant. There are two indications, one by subtraction, the other anecdotal (and ugly).
Ruckleshaus convened a scientific panel under Judge Sweeney to study the merit of banning DDT. The Judge submitted a 9,000 page report of his findings on DDT:
There was clearly no real environmental motive for banning DDT. There are numerous possible motives on the part of EDF's financial supporters, from preventing third world development to maintain cheap access to minerals, pesticide companies wishing to market organophosphates, political interests in maintaining third whirled dependency upon American food and financial aid, deepening the control of the IMF, to third world population control. It is my opinion that the latter was the overriding motive.
When confronted with the evidence and the 270,000,000 cases of malaria and the millions of lives lost every year to infections that could have been prevented, Dr. Charles Wurster, chief scientist of the Environmental Defense Fund (and not at all qualified by specialty for his position) is quoted to have stated, "this was as good a way for getting rid of (overpopulation) as any".
The quote has not been authenticated, although it has been published numerous times. Neither Dr. Wurster nor the notoriously litigious EDF has sued a publisher of the quote.
Finally, how did they pull it off?
I'm not sure what you mean by this question, politically or legally? Under what authority, you might ask?
Politically, the case is obvious: control of the media buzz. Legally I believe the commerce clause, NEPA, and (IIRC) the Clean Water Act are cited for the Secretary's authority on such matters, but that is all from very deep memory off the top of my head.