Most large corporations routinely require the same fair and equal treatment of people in their internal corporate rules and regs. Passing such ordinances was seen as making those communities more attractive to potential employers who could offer the state - and the cities - well paying jobs.
Outside those municipalities, any kind of discrimination against LGBT people was and is perfectly legal. Further, state law outweighs municipal laws, so any law suit under those ordinances has at best limited use.
What the advocates for these laws claim as religious freedom is false, just as you cannot discriminate against people on the basis of race claiming religious belief. There were multiple law suits that have established that, for example in the Bob Jones University cases where discrimination was not allowed against black people, or against mixed race married couples or mixed race dating.
The arguments that are being presented, that Jewish or Muslim restaurant owners could be forced to serve pork sandwiches are false, so false they can fairly be called lies. No one is now or has ever under the guise of anti-discrimination forced a business to provide a product or service. What anti-discrimination has required is that if you sell something (a product or service) to others, you can't refuse to do so to a group of people because of the religion, race, gender, etc. of that group.
So no one is going to force any bakery to bake penis shaped cake --- but under anti-discrimination laws, if they DO make such cakes for heterosexual customers (like bachelorette parties) THEN they have to sell them to a gay customer too. If you sell wedding photography to straight couples, then you have to do so to gay couples.
The one argument that is made in support of such legislation that is valid is that a photographer who provides their services to events could be required to provide them to an event with which they do not agree. A parallel example of how that is in fact the very definition of freedom is that, for example when the National Socialist Party of America aka American Nazis versus Skokie, IL, Nazis had as much right as any other group to freedom of assembly in parading through a city where many Jewish Holocaust survivors lived. The SCOTUS correctly decided that special protection or accommodation could be made for those people on the basis of religion, while the Nazis, however repellent they were as human beings, did have the right to freedom of assembly in public places --- and ONLY in public places.
However much radical religious right extremists wish it to be so, corporations or other businesses are not people. PEOPLE have individual liberties; other entities have some rights as entities which are legal fictions, but they are not people. If you are a person doing business as a business entity, then you have to abide by the laws regulating businesses, which does not include religious discrimination.
But, by the same token, political beliefs by a group are not a protected area of activity. You can say HELL NO to a bunch of KKKluckers [members of the Klu(eless) Klux Klan] just because you don't like the glare coming off their ugly little bald skin-heads or because you object to their stupid little recycled bedsheet outfits. You don't need the protection of freedom of religion to reject them as clients.
Likewise, there is not instance where anti-discrimination could result in someone being rounded up and forced to march in a gay pride parade. If you are a bigot, you can't prevent such a parade, the same way aggrieved Jewish Holocaust survivors couldn't prevent a rally or a parade by offensive Nazis. In fact, under freedom of assembly and freedom of speech, you can protest at such an event.
If you don't like selling gay wedding cakes (penis optional) or taking photos at a gay wedding, get out of the business. You aren't protected by the Constitution from serving clients you don't like on the basis of your religion, just like you can't refuse to seat Jewish or Muslim people in your restaurant because you are a disgusting and reprehensible bigot --- so what?
Conservatives seem to have a pattern of desiring to discriminate against other people, while claiming, falsely, and usually in an annoyingly whiny way, that they are victims themselves, when they are not.
Which prompts the inevitable question -- why do conservatives persist in defining their freedom in terms of denying freedom and dignity to others, over and over and over? Shame, shame on them. Why as well, do conservatives persist in lying, and in trying to scare people when there is no reason for fear?
They are just silly, and perhaps they are so used to projecting their own lies on others, they can't believe anyone else is a better human being than they are. They assume the worst, and they are wrong.
Shame on Arizona, shame on all the other states trying to legislate religious prejudice and bigotry, shame on all the states trying to promote radical right wingers taking away liberty from people. THEY are UN-AMERICAN.