It's tricky because dissenting doesn't necessarily mean you'd reach the opposite conclusion in every respect.
In this case, I really doubt even the most conservative justices believe "birthright citizenship means whatever an Executive Order says it does." At a minimum, we know they aren't signing on to the reasoning the 5 in the majority used. And then we can learn whatever they feel like saying in the dissent, but a dissent is just an essay with no force of law.
But since we have the dissent opinions, it's not tricky at all to see why they are dissenting.
Alito and Thomas straight up believe that the constitution does not provide birthright citizenship and the executive order is valid. Gorsuch mostly agrees but makes an exception if the parents plan to stay in the US. Kavanaugh agrees that birthright citizenship is not provided by the constitution, instead he argues its a federal statute that congress can overturn (but the president cannot)
Thomas and Alito complained that they were being impeded from setting executive policy. Very easy to understand the argument that this court is too small.
The joke that Supreme Court judges are high priests interpreting the constitution as if it was holy scripture was in fact accurate description all along
> The Court today takes the extraordinary step of holding
facially unconstitutional the President’s Order excluding
from citizenship the children of foreign temporary visitors
and illegal aliens.
In this case, I really doubt even the most conservative justices believe "birthright citizenship means whatever an Executive Order says it does." At a minimum, we know they aren't signing on to the reasoning the 5 in the majority used. And then we can learn whatever they feel like saying in the dissent, but a dissent is just an essay with no force of law.