Can you explain to me how baking a cake for someone infringes on your rights?
If you need to have it explained, it's not likely that anyone could explain it to you. It's so incredibly obvious that if an explanation is required, then that same explanation probably won't be understood.
Some of that has to do with the fact that a lot of people haven't really thought through the whole concept of "rights". It's one of those terms that we use a lot, but which a lot of people haven't thought much about. It gets into the whole question of "legal rights" versus "natural rights", and in the US constitution it gets even trickier because the Constitution defines a sort of two layer system of legal rights, in which some rights can be abridged by the legislature, while others cannot.
And then even that gets messed up, because the Supreme Court gets involved and declares that some practice or other gets declared unconstitutional. The effect is that there is a brand new legal right which people enjoy today, but did not enjoy in years past, except that in order to make it fit in with the legal theory of the constitution, the only way they can create the new right is to insist that the right existed for a long time, but no one had really noticed it until the court ruled in a specific case.
If I am quite honest it just makes the bakery sound dead ass petty.
In which case, if you outlaw the practice, you are infringing on their right to be dead ass petty. Feel free to do so, but recognize that you are restricting their legal rights.
Any time you make something illegal that had previously been legal, you are infringing on someone's legal rights. That just a definition. In ages past, a restaurant owner had the right to refuse service to black people. Today, they don't. Those discriminatory restaurant owners have lost a legal right.
Today, some people say that cake bakers should have the right to decline to bake cakes if the message on the cake is offensive to their religious beliefs, and they also note that the cake itself carries a message. Other people say that although the cake baker should be able to decline some sorts of business offensive to his religious beliefs, for example refusing to make Halloween themed cakes, there are other messages that are so important that the cake baker should be compelled to create such cakes as a condition of doing business. Of such fine distinctions are court cases made.
But we are getting away from the OP. The question of whether two people ought to be allowed to enter into a legally defined, state-sanctioned, partnership is separate from the question of who ought to be required to bake a cake to celebrate the occasion, and both of those questions are at best peripheral to the question of what is the conservative mindset that explains how the laws create special rights for gay people. If you are interested in expanding the OP to include the question of the conservative mindset that supports allowing people to continue to operate bakeries, even though they won't bake cakes for gay weddings, that's one that I actually understand. I don't understand why the right to get married is a "special right", and I would call it a stretch to say that the right to buy wedding cake is a "special right", but I do understand why some people would object to closing down bakeries who refuse to make cakes for gay weddings.