First, I want to point out that Arizona is not a barren wasteland. There are green and mountainous areas near Flagstaff, and lots of nice areas like Tucson and Sedona.

But as for why European and US settlers went to Arizona, there are a few reasons.

The first Spanish in the regions were conquistadores and Catholic missionaries, who went there to look for gold or explore generally, and to convert the natives.

The US built forts and settlements in Arizona during the 1800s, after it was taken from Mexico, in order to maintain control of the territory and make safe land routes for settlers going west. Even then, there was cattle ranching and agriculture. Fun fact: Arizona territory supported the South during the Civil War.

Later, Mormons travelled to the territory (and to other desert regions) to set up Mormon settlements far from persecution by Christians or regulation by the US government.

Other people settled there from the 1800s to the early 1900s to do agriculture, ranching, and mining. "Deserts" are really the best for farming if you can build canals for water (which was done by the native Americans in Arizona since like 1000 years ago). Citrus and cotton both grow great there, and cattle grazing is good too, making up a $10 billion dollar industry today.

As for mining, Arizona produces like 60% of copper mined in the US since the beginning of the 1900s. They mine lots of other minerals there too. Many of the towns that popped up in AZ from the 1880s to the 1920s were built on mining claims. They also had a gold rush in the 1850s and 1860s.

After the invention of air conditioning in the 40s and 50s, lots of retirees moved to AZ for the warm climate and cheap cost of living, so now a lot of the state economy is in health care and related industries.

Anyways, my parting point is that 1) Arizona is not all desert, and 2) Contrary to what people often think, deserts are great for farming and grazing if you can run water there. Many of the best farming regions in the world are hot and dry.

It is worth noting that the Phoenix area hasn't always been as hot as it is now. Some of the heat is due to the so-called "urban heat island" effect, where urban areas are simply hotter on average: this is due to the clearing of vegetation (hence less cooling by evapotranspiration and less tree cover) and the use of materials like concrete and asphalt that have fairly high heat capacities (this heat is released at night, so while a city might be something like 5 degrees Celsius hotter than the surrounding area during the day, this can be more like 10 degrees Celsius at night). Buildings also provide multiple surfaces for heat to reflect off of and block wind flow, amplifying the heating effect.

It can be hard to quantify the exact magnitude of a heat island effect, but this paper suggests that some parts of the outlying Phoenix suburbs, which have been developed greatly during the last fifteen years or so, have heated by up to 0.5 degrees Celsius per year (or a total of 7 degrees Celsius during the years 2000-2014). This paper found more modest increases, on the order of 2 degrees Celsius, during a slightly earlier time window (1990-2004), though it isn't as fine grained. This paper is a little older and examined winter temperatures, finding increases on the order of 2-3 degrees between 1920-ish and 1977-ish. So the heat island effect continues to increase mean temperatures, and in the past the Phoenix area, though very warm, would not have been as hellishly hot as it is today.