I can't speak for lawmakers in Arizona, but I do know that many people, especially people in some of the states that border Mexico, are extremely frustrated at the way in which the U.S. government enforces or doesn't enforce immigration laws. I consider it possible that the people "Arizona" hopes to provoke are the people in Washington who could reform immigration policy.
Thanks to Rex for providing the information about the law.
As for the implementation, it is problematic at best. As Katydid pointed out, many Americans do not carry identification that would prove their legality. For example, the other day I went for a bike ride and debated about taking my wallet (containing my driver's license) with me. In the end I did, but many people probably wouldn't. I do have my voter I.D. card in my wallet, which proves my citizenship, but many citizens aren't registered voters. I only carry my passport when traveling out of the country. Again, many citizens don't have a passport.
As others have pointed out, it is impossible to judge from skin color or English skills whether someone is a citizen or legal resident. I know people who would be considered highly suspect were they to be in Arizona. Despite their national origin and accented English, they are citizens or long-time legal residents.
Whatever the intentions of the lawmakers, it is nearly certain that some citizens will be wrongly suspected, unable to prove their status and wrongly detained. I am reminded of a time I went to the movie theater when I was 14. I was part of a group of girls from my class, some of whom were older than I. The oldest was 16, had a driver's license and had, in fact, driven us to Des Moines. My very short, petite friend bought her ticket first, asking for the reduced "14 and under" ticket price. Others asked for "14 and under". No problem. When it was my turn, the cashier frowned. She didn't believe I was 14 and asked me for I.D. I stood there with my mouth open. To begin with, I was insulted: I wouldn't have lied. Second of all, despite being the tallest in the group, I was the youngest, and really was 14. Exactly what I.D. did the cashier think I would have on me? How did she expect me to "prove" that I was a kid? She did eventually let me buy the 14 and under ticket, but she also sold one--without batting an eyelash--to the 16-year-old driver, who happened to the second shortest in the group.
I'm pretty sure things will work the same way in Arizona. No I.D.? Dark skin? Spanish accent? You'll get hauled off. Undocumented immigrant from Poland? You may not even be asked for your I.D. (Please note, I have nothing against Poles. I just happen to know a Pole who came to the U.S. on a short-term visa with the goal of ditching his job as dishwasher at a church camp, traveling to Chicago, disappearing into the immigrant community and living there illegally. Within a week of arriving in Michigan, he did just that.)