I was just at Chaco Canyon a couple weeks ago!
I've been to Mesa Verde twice, about 8 years apart. The stories I heard both times were different, based just on what archaeologists had learned in the meanwhile. At first I was told that living in between these mesas depended greatly on 'resource management', i.e. collecting what water they could and storing it carefully, hunting/gathering over a large area, etc. And they were called 'Anasazi' then--not the name of a tribe but the word the Navajos used for them, which actually meant 'ancient enemies' or 'ancient nemesis'.
But the current thinking is that this area wasn't a harsh desert in those days, that the 'Mesa Verdians' and 'Chacoans' (as they're called now) could do 'dry land farming'. It took more land, but they had plenty of that. Apparently they had plenty of water (perhaps they used it more carefully than we do). At Chaco Canyon they even had a dam across the stream. Apparently defense was important also because they built in places that were very defensible.
If you look at the ruins you can see that they had a lot of free time! The biggest building at Chaco, now called Pueblo Bonito, had 600 rooms! It was the biggest building on earth up to the 1880s. It is thought to have taken 200-300 years to build! That suggests a well-ordered, stable society that can work on a project for that long, with no terrible shortages of food or water, and no major wars.
They all picked up and left around 1100-1200 AD, it's thought because of a prolonged drought.