"It is now clear that the challenged laws burden the liberty of same-sex couples, and it must be further acknowledged that they abridge central precepts of equality . . . Especially against a long history of disapproval of their relationships, this denial to same-sex couples of the right to marry works a grave and continuing harm. The imposition of this disability on gays and lesbians serves to disrespect and subordinate them. And the Equal Protection Clause, like the Due Process Clause, prohibits this unjustified infringement of the fundamental right to marry." (page 22, from the coverage on SCOTUSBlog)
For others, the opinion also reasserted that people who are really mad about this can continue to be mad and vocal about it, as guaranteed by the First Amendment.
"Finally, it must be emphasized that religions, and those who adhere to religious doctrines, may continue to advocate with utmost, sincere conviction that, by divine precepts, same-sex marriage should not be condoned. The First Amendment ensures that religious organizations and persons are given proper protection as they seek to teach the principles that are so fulfilling and so central to their lives and faiths, and to their own deep aspirations to continue the family structure they have long revered. The same is true of those who oppose same-sex marriage for other reasons. In turn, those who believe allowing same-sex marriage is proper or indeed essential, whether as a matter of religious conviction or secular belief, may engage those who disagree with their view in an open and searching debate." (page ~32)
The full majority opinion [PDF]: http://www.supremecourt.gov/opinions/14pdf/14-556_3204.pdf