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Ignorance is a word for the universe before it developed Mind with its ability to discriminate, recognize, judge, separate, and set goals.
Most of the Twelve Nidanas describe development and taking shape of the above qualities and corresponding representational experiences.
The naive mind, as it develops, delineates entities, calls some of them desirable, designates the logical center of experience as "self" and engages into purposefull action to get this self in position to acquire the desirable entities (and other experiences).
Rebirth refers to the tendency of this process to cyclically recreate itself despite the limitations of physical media.
"Birth" specifically refers to the birth of "self" as the designated center of experience and action. The way the birth of self depends on craving for entities and on the purposefull action, is described in the Twelve Nidanas. Basically, our perception of distinct entities develops in conjunction with taking shape of our like/dislike attitude which eventually matures into Self based on the mental fabrications we engage into on the way to acquiring capacity for purposeful action towards the desirable entities.
It's not really an anthropocentric view, it is representation-centric and information-centric, with media playing secondary role.
Uprooting ignorance refers to clearly understanding (and seeing in one's own real life) how this process works. It has an effect of dispelling the illusion of solidity that depends on not understanding the mechanisms behind the magic trick.
To say that rebirth has ended is a simplification though, because technically the process at large still goes on, you just no longer identify with any part of it. So for you it's the last birth because now you know there was no you ever, to begin with. But for others it still continues. The tendencies that led to your Liberation/Enlightenment do not completely disappear though, which is why in Mahayana we say that enlightened Bodhisattvas keep on existing in some sense, helping new students from the realm of subtle elements.
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Some context is required, suggest you read about Dependent Origination: Learning materials for Dependent Origination (Paṭiccasamuppāda) in Theravada Buddhism
Long story short, what we know about the origin of Life is too limited. The Universe doesn't "care", it is just how things are. Lord Buddha mentioned he has only taught us a few things compared to the grand scheme of things he knew.
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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?
They don't, but humans do. At least those who care enough about ending Dukkha.
Why does uprooting ignorance and craving stop the rebirth process?
Because human existence, like any other natural systems, needs some kind of fuel to keep the wheel spinning. For ours, the fuel is ignorance and craving. That's why "uprooting" is a particularly fitting word. The tree once "uprooted" will not grow back again.
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"Rebirth" is a moral teaching about the results of kamma. It means the idea of "self" reappears again; attached to the results of an action. "Becoming" is an "asava" or "mental defilement". It does not mean "reincarnation". Uprooting ignorance and craving stops the rebirth process because it stops the "birth" of the "self" idea.
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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.