What is the role of karma in rebirth? Rebirth is not reincarnation.  How then does the karma of an individual affect their rebirth?

In reincarnation, an individual's karma results in a better or worse next life for their soul. But in rebirth, there is no soul. So why should we care about our karma regarding our rebirth?
But in rebirth, there is no soul


The Buddha didn't say there was no soul. 

The problem with saying there is a self or there is "no self" requires a view which the Buddha said was incompatible with the practice:


  ... a certain monk said to the Blessed One: "Which birth, lord? And whose is this birth?"
  
  "Not a valid question," the Blessed One said.
  If one were to ask, 'Which birth? And whose is this birth?'
  and if one were to ask,
  'Is birth one thing, and is this the birth of someone/something else?'
  both of them would have the same meaning,
  even though their words would differ.
  When there is the view that the soul is the same as the body, there isn't the leading of the holy life.
  And when there is the view that the soul is one thing and the body another, there isn't the leading of the holy life.
  Avoiding these two extremes, the Tathagata points out the Dhamma in between: From becoming as a requisite condition comes birth."
  
  SN 12.35


He didn't say the one thing (there is a self) or the other thing (there is no self), he said the thing in the middle, which is that with certain phenomena as a requisite condition, other phenomena arise.

He didn't answer the big question of what does and does not exist, he answered the small question about how a thing arises.

Some reading: No-self or Not-self?


  So why should we care about our karma regarding our rebirth


One thing that the Buddha consistently affirmed, though, is the efficacy of action and the results of kamma. He would go out of his way (MN 101) to refute ideas that personal action had no consequences or was totally predetermined.

He just didn't link together action and an individual's identity through multiple lifetimes or say that they weren't linked. Either view prevents the living of the holy life.

Seeing action as efficacious and producing results is the thing that is necessary; understanding the arising of phenomena is necessary. Talking about souls is not necessary.