See section II B of the opinion. On page 13:
On July 25, 2014, the Commission met again. This meeting, too, was conducted in public and on the record. On this occasion another commissioner made specific reference to the previous meeting’s discussion but said far more to disparage Phillips’ beliefs. The commissioner stated:
“I would also like to reiterate what we said in the hearing or the last meeting. Freedom of religion and religion has been used to justify all kinds of discrimination throughout history, whether it be slavery, whether it be the holocaust, whether it be—I mean, we—we can list hundreds of situations where freedom of religion has been used to justify discrimination. And to me it is one of the most despicable pieces of rhetoric that people can use to—to use their religion to hurt others.” Tr. 11–12.
The analysis of whether it was really that bad follows, ending on page 16. The first paragraph of that analysis describes the reasoning quite well, however:
To describe a man’s faith as “one of the most despicable pieces of rhetoric that people can use” is to disparage his religion in at least two distinct ways: by describing it as despicable, and also by characterizing it as merely rhetorical—something insubstantial and even insincere. The commissioner even went so far as to compare Phillips’ invocation of his sincerely held religious beliefs to defenses of slavery and the Holocaust. This sentiment is inappropriate for a Commission charged with the solemn responsibility of fair and neutral enforcement of Colorado’s anti-discrimination law—a law that protects discrimination on the basis of religion as well as sexual orientation.
The analysis also considers that the Commission's reasoning in this case was inconsistent with its reasoning in "the cases of other bakers who objected to a requested cake on the basis of conscience and prevailed before the Commission." Those cases concerned bakers refusing orders for "cakes depicting anti-gay marriage symbolism." It then says that the ruling of the Colorado Court of Appeals, which upheld the Commission's ruling as permissible "because of the offensive nature of the requested message," was improper because "it is not, as the [Supreme] Court has repeatedly held, the role of the State or its officials to prescribe what shall be offensive."