What does the cross section of the trunk of the tree show
The rings show the age. One ring per year in most cases. The SW US sometimes adds two rings per year, depending upon the weather that year. The distance between the rings indicates the weather that year. A wide ring is a wet year with a longer growing season. A narrow ring means drought year. The pattern if wide/narrow may also show some of the history of the local site. If the center rings are all narrow for many consecutive years then suddenly get wide then something happened. The cluster of narrow rings could indicate the tree was suppressed, growing underneath a larger tree that was blocking sunlight and out-competing for water and nutrients. Then the larger tree died and the smaller tree was now free to grow fast.
Old injuries may show up in the cross section. Old scars get grown over by subsequent years. Fire history studies are done this way. The burn scar may still be visible in the cross section. Counting the rings back from when the tree was cut reveals when the fire occurred, and sometimes approximately when during the year it occurred (early summer, late summer, fall). By looking at tree scars over a larger area you can determine the size and extent of the fire.
You do not have to cut down a tree to determine these things. A tool called an increment borer removes a small core sample without harming the tree.