Why do ignorance and intention have something to do with rebirth? I have a lot of trouble dealing with rebirth, mostly because of my previous "skeptic" past. However, buddhism has helped me a lot to open my mind and to get detached from any point of view that might be incompatible with the wisdom shown by the practice of the Dhamma.

This aperture to new perspectives has made me reconsider the possibility of rebirth. Thus, my question.

From the point of view of paticcasamuppada, ignorance and craving are prerequisites for rebecoming. 

But why would the universe and its underlying mechanisms and laws care about our ignorance, intentions and desires? 
Isn't this a kind of anthopocentric view?
Why does uprooting ignorance and craving stop the rebirth process?
Let me introduce you to the old South Indian Monkey Trap (from this article):


  In Zen And The Art Of Motorcycle Maintenance, Robert Pirsig’s
  bonkers-but-brilliant philosophical novel that turns 40 this year, he
  describes “the old South Indian Monkey Trap”. ... The trap
  “consists of a hollowed-out coconut, chained to a stake. The coconut
  has some rice inside which can be grabbed through a small hole”. The
  monkey’s hand fits through the hole, but his clenched fist can’t fit
  back out. “The monkey is suddenly trapped.” But not by anything
  physical. He’s trapped by an idea, unable to see that a principle that
  served him well – “when you see rice, hold on tight!” – has become
  lethal.


The monkey needs to let go of the rice in order to free himself from his suffering. The way to end his suffering, is to end his craving for rice. He got stuck in the trap in the first place due to his craving for rice.

But in order to end his craving for rice, he must first understand how his hand is stuck inside the coconut. When the monkey overcomes his ignorance about how the trap works, he would let go of his craving for rice, and release his clenched fist. With this, he would be free from his suffering.


  OP: But why would the universe and its underlying mechanisms and laws care about our ignorance, intentions and desires? Isn't this a
  kind of anthropocentric view? Why does uprooting ignorance and craving
  stop the rebirth process?


Back to my analogy, does the coconut or the rice care about the monkey and its intentions, desires and ignorance? Not at all. In no way is this a anthropocentric view.

Rather, as Andrei stated in his answer, it's an information-centric or representation-centric view. I also think of it as a conceptuality-centric view. The suffering and its release are based on the mental concepts or ideas of the monkey with regards to its identity, desires and relationship to its environment. 


(Illustration above: Paul Thurlby for the Guardian)