The argument is that legacy admissions are a criterion being used in lieu of race to achieve racial discrimination. Similar systems were an aggressive part of the Jim Crowe South to prevent blacks from voting when they couldn't make skin color-specific rules anymore, where you would have to prove literacy or your grandfather's ability to vote to get a vote in since they knew that no black person could pass those tests.
It's not the strongest case, but it isn't unreasonable and it has a rational basis. I'm skeptical that even a disinterested court would accept the logic on its face and know that the current court will just wallow in its hypocrisy, but I at least understand the principle.
The defense to it is that these elite universities actively desire diversity, so clearly they are not using legacy admits as a proxy for racial discrimination. There's plenty of evidence that there is a different, corrupt-but-legal purpose for these legacy systems that is not racially motivated.
edit: On the flip-side, if indirectly discriminatory policies like these ARE allowed, it gives the universities a clear path forward to continuations of their AA policies. They just need to replace the race of their applicants with proxies for race that are allowed. Theoretically, this logic could force a standoff -- either policies that indirectly have discriminatory outcomes (defined by who?) are illegal or they aren't. Or else you'd have to somehow prove intention to discriminate before you could find a policy illegal, which is another very complex can of worms that the elite universities could probably get around.
And so, like with everything around this case, it likely won't change any real behavior from the elite universities but will seriously fuck with less affluent ones.
ganksy