https://www.wnycstudios.org/podcasts/radiolab/articles/alpha... and https://www.wnycstudios.org/podcasts/radiolab/articles/retur... are good listens on the subject.
In particular from the second one:
> SOREN: Which brings me back to that little part of the article that piqued my interest in the beginning, because the pig that they got the kidney from to do this was a very special pig. Normally, pigs and other mammals that aren't primates have a sugar in their body that our bodies don't have, and so we don't like it or see it as foreign. And that's why usually an organ from another animal would get rejected. But this pig had been genetically modified. It had had the gene that makes that sugar removed, so it didn't have that sugar. Which is part of the reason this worked. And that sugar just so happens to be called "alpha-gal."