The brief is here. As you might expect, it is written largely for Justice Kennedy. I was interested to see that the brief made two arguments that I personally think provide the strongest constitutional defenses of Prop 8: first, that there is a rational basis in maintaining the status quo for now in the face of rapid social change (pages 48-55 in the brief; my related blog posts on that argument are here and here); and second, that there is a rational basis in supporting Prop 8 to express opposition to perceived judicial overreach (pages 55-61 in the brief; my blog post on that argument is here).
I find it hard to predict what the Court might do in this case: If it reaches the merits, I can see it going either way. Anyway, stay tuned for the brief of the Respondents, which will be filed next.