I'll explain it as I understand it. (And hope it's correct, if somewhat simplified...)
There are two kinds of Coombs test:
The direct Coombs test and the indirect Coombs test.
The indirect Coombs test is done on pregnant women's blood. It is what is called the antibodies test. If it is positive, it means that the woman's blood has developed antibodies against the Rhesus factor or some other factor that crossed into her blood from her unborn baby's blood. She might also have got these antibodies from a previous pregnancy or birth or from some other cause altogether.
Let's pause quickly to explain what the Rhesus factor is:
The Rhesus factor is something that exists on the surface of red blood
cells. Being Rhesus factor positive means one does have the Rhesus
factor. Being Rhesus factor negative means that one does not have the Rhesus
During pregnancy and more so at birth, a problem can occur if the mother is Rhesus factor negative and the child Rhesus factor positive. If in such a constellation the bloods of mother and baby mix, which can very rarely happen during pregnancy or somewhat less rarely at birth (although it is not supposed to happen normally), then the mother will make antibodies against the child's Rhesus factor, which is alien to her blood.
It takes some time for these antibodies to form, so they would not harm this particular child. However, if her body continues to produce these antibodies, they can then harm a future child. Because in a future pregnancy, these antibodies against the Rhesus factor can cross the placental barrier and get into the unborn child's blood. If the unborn child is Rhesus factor negative, no harm will happen. But if the child is Rhesus factor positive, then these antibodies attack the Rhesus factor. As the Rhesus factor is on the red blood cells, what this basically means is that the red blood cells are attacked. The red blood cells are in charge of transporting oxygen through the body. If they don't work properly, then the unborn child has anemia. As a result, the heart will try to compensate by pumping the blood quicker. This over-exertion of the heart causes fluid to accumulate in the body (e.g. the heart and lungs) and in the worst case, it can lead to heart failure and thereby stillbirth. In a mild case, the unborn baby might be completely fine. After the birth, such babies might possibly still be completely fine or just show more pronounced jaundice.
Jaundice occurs as a result of the breakdown of red blood cells. One by-product of the breakdown is bilirubin, which causes the yellow colour.
In a bad case, a Rhesus positive baby whose red blood cells are being broken down by maternal antibodies, would have very severe jaundice and even require an exchange transfusion, whereby the old red blood cells are removed and replaced by new ones (from donor blood).
The name for the disease that occurs when a Rhesus positive child's red blood cells are broken down by maternal antibodies is hemolytic disease of the newborn or Rh (Rhesus factor) disease.
Remember I previously mentioned there are two types of Coombs test, the direct and the indirect one? Here the direct one comes in. It is done with cord blood after the birth, if mother is Rhesus negative. If the direct Coombs test is positive, it means baby has hemolytic disease of the newborn.
This disease was more common before. (Although it by no means happened to a majority of Rhesus negative women, as is sometimes suggested. Even back then, it was among the rarer things to happen.) Nowadays it is extremely rare, because Rhesus negative mothers who give birth to Rhesus positive babies receive a shot of anti-D within 72 hours after the birth. Often Rhesus negative mothers even already get one or two shots in pregnancy, either routinely or after an accident or bleeding. This might be helpful, but is not as relevant as the shot after the birth, which is much more important, as it is at birth that the bloods of mother and child are most likely to get in contact, if at all.
The anti-D shot contains antibodies against the Rhesus factor. As they are provided from an external source (the shot), the mother's body does not need to produce them herself and therefore has no memory for producing them, meaning it won't produce them. Therefore no risk for future babies. (In very very rare cases, the shot might not work and a mother might produce her own antibodies anyway.)
So to sum the whole story up and relate it to your Aunt, from your story it seems like your Aunt, who is supposedly Rhesus negative, developed antibodies against the Rhesus factor, which caused hemolytic disease to her fourth and fifth baby, who both must have been Rhesus positive.
Sad story. I hope she and her family are doing fine nowadays?
My heart-felt regards to her and to you for trying to understand what happened. I hope you understood my explanations. And to you others out there who understand these things better than me, if I said anything incorrect, please correct me.