I don't see why people care whether someone is muggle-born or not. Does it make any difference? Please illuminate me, I don't understand.
It's magical racism, basically. A reason to consider others below your station. The worst of the pure-bloods are akin to Nazis in there desire to keep Muggles or Mudbloods in their place. That place being servitude or even death.
This statue at the Ministry of Magic depicts Muggles being crushed.
A lot of wizards think muggles are inferior, because they are (of course) incapable of magic.
So there are wizards who think that wizards who are born to muggle parents (mudbloods) cannot become so good at magic as pure blood wizards (wizards who don't have muggle ancestors). They think mixed marriages between muggles and wizards are "thinning" or "contaminating" wizard blood and are detrimental to the wizarding world. However, they are wrong, because (for example) Hermione is very capable in magic, despite having only muggles as parents.
Some aspects of human genetics (such as height) are complex and poorly understood, but generally tall parents have tall children and short parents have short children but there are constant exceptions. Some aspects are even less well understood, such as human intelligence. Some aspects of human genetics are simpler and now better understood, such as certain forms of color-blindness.
Magical society, on the other hand, does not have scientists, or even science (as we know it) and so many of the behaviors and expectations of wizards and witches are shaped by the vague preconceptions that human society has evolved where it comes to genetics.
Whether or not a person has any magical ability appears to have a strong (though abnormal) genetic basis in that most muggles have muggle children and most wizards have wizard children. The abnormal part is that if magical ability were a recessive gene, then marriages with mixed muggle/wizard parents would produce children who had a 50/50 shot, at best, of being wizards. Instead, children of mixed marriages are almost all wizards.
Note that if magical ability were a dominant gene, it would not be possible for two muggles to have a wizard child.
There is some evidence in the novels that strong magical ability does flow through certain genetic lines (Voldemort is a descendant of Slytherin after all), but on the other hand, there is no compelling evidence that this ability is either diluted when mixed with non-magical blood or strengthened by pure-blood unions.
The bottom line though is that prejudice against muggle-borns appears to be pure prejudice based on a traditions that are without merit. The strength of one's magical ability, like many human abilities, is more a matter of interest and drive than of genetics.