What exactly are the last two of the Arupa Jhanas? How can we transcend them and for what? In the words of Buddha, the world in which we are living is generated as the following:


  From ignorance as a requisite condition come fabrications.
  From fabrications as a requisite condition comes consciousness.
  From consciousness as a requisite condition comes name-&amp;-form...  


And, additionally, the following loop is also expounded: 


  "From name-and-form as a requisite condition comes consciousness. From
  consciousness as a requisite condition comes name-and-form..."


The full understanding (awakening) of this knowledge is given through the eight jhanas (particularly the last four Arupa Jhanas;) which relates to what was said precedingly as follows: 

From the complete transcending of each following states, the bhikku enters &amp; remains in the dimension of:


Infinite space
Infinite consciousness
Nothingness, then at last
Neither-Perception-Nor-Nonperception.


My question will be about the last two. What are they exactly, the Sphere of Nothingness and the Sphere of Neither-Perception-Nor-Nonperception? How can they be transcended, and what for?
Nothingness is transcended by seeing the relative coarseness of perception even of nothingness. By observing the perception repeatedly, one sinks into a state of quasi-perception. As per the Vism:


  When, however, he wants to develop the base consisting of
  neitherperception nor non-perception, he must first achieve mastery in
  the five ways in the attainment of the base consisting of nothingness.
  Then he should see the danger in the base consisting of nothingness
  and the advantage in what is superior to it in this way: “This
  attainment has the base consisting of boundless consciousness as its
  near enemy, and it is not as peaceful as the base consisting of
  neither perception nor non-perception,” or in this way: “Perception is
  a boil, perception is a dart … this is peaceful, this is sublime, that
  is to say, neither perception nor non-perception” (M II 231). So
  having ended his attachment to the base consisting of nothingness, he
  should give attention to the base consisting of neither perception non
  non-perception as peaceful. He should advert again and again to that
  attainment of the base consisting of nothingness that has occurred
  making non-existence its object, adverting to it as “peaceful,
  peaceful,” and he should give his attention to it, review it and
  strike at it with thought and applied thought.
  
  Path of Purification X.40 


Transcending the sphere of neither perception nor non perception is not directly accomplished in the same way as the others; it is considered the height of tranquility meditation, there is no further one can go with that type of practice. To transcend it means to switch to insight meditation; since insight meditation is considered higher than tranquility, it can be seen as the "next step", even though it is in a new category of practice and attainment.