Side note: the cake maker was happy to provide the service until told the cake was for a lesbian couple.
Finding something offensive and refusing to participate in something offensive is your right only so far as the law recognizes that right. As an individual, having to pay taxes may be offensive, but the individual is required to pay taxes regardless of beliefs. As a business, refusing to offer a service the individual may find offensive is illegal if the service is refused based on the customer's protected class.
A cake maker may refuse to create a cake for anyone based on any personal preference, unless that preference is contrary to the law. In some states, it is illegal to refuse service to someone based on their sexual preference.
Personally, I find the compulsion to offer a service you believe is immoral to be objectionable, but I do understand the need lawmakers feel to protect certain classes from discrimination. With race and sex, the individual has no choice, and so discrimination against women, men, blacks or whites seems like something that the law should consider. However, with religion and sexual preference, the only way that someone can be a part of a protected class is by making a decision. One could argue that you are a part of a religion even by abstaining from participating any religious activity since "atheist" and "agnostic" are labels used to describe those belief systems. Even so, I have no personal objection if a Muslim refuses to create cakes for Christian celebrations. With sexual preference, the labels are far more fluid. In that particular class, the only way a person can be part of that protected class is to choose to participate in an activity they could abstain from.
The law, in some states, requires businesses to provide service to a protected class which is solely based on the choice made by members of that protected class to choose to be in that class.
Bluntly put, I can choose to abstain from sex but I can't choose to abstain from being black. I feel the law should protect me from people who would discriminate against me based on my skin color, but the fact that I'm a virgin with no intention of ever having sex shouldn't keep me from being given rights people who make a different choice are given.