They look at tree rings. Tree rings are created as trees grow each season, and depending on the temperature they can be thicker or thinner. You can look at a slide of tree and work backwards from now to the birth of the tree years ago and measure the relative difference in temperature. To go back further, you find old wood from a tree whos life overlapped that of the first tree and you line up the growth rings of similar thickness so you can extend the pattern back. Keep doing that with older and older samples and you have a timeline going back for really long periods.
There are only two problems. First, tree ring thickness is not affected by temperature alone. Other atmospheric factors affect it too and there is no way to know which ones were really driving a particular pattern. Second, rather than using a large sample size of many trees to construct the timeline, the scientists cherry picked a small handful of trees that supported their data. In fact the famed "hockey stick" is all based on just ONE tree from Siberia.
So basically the use of tree rings has inherent limitations and can be manipulated by using small sample sizes and carefully choosing which samples to include (which was how the post 1950 temp surge was created). Other than that, it's a great idea.
Given all of the variables that can affect tree growth, I seriously doubt that any estimate of temperature made on the basis of tree ring measurements would have an error of less than a few degrees.
It certainly does not have the accuracy of a digital thermometer that can measure 1/100 degrees, or even an alcohol (or mercury) thermometer than can measure to 1/10 of a degree.
I have been growing trees in my orchard for quit a few years now, the only thing that seems to effect the size of the rings is availability of water.
More water, wider ring; less water, narrower ring.
Really cold summer (less than 85 all summer) but plenty of water, wide ring.
Really hot summer (110 to 115 all summer) but plenty of water, wide ring.
Should have said; Spring, Summer and Fall, winter doesn’t count, because all my trees are deciduous and dormant in the winter.
I think dendrochronology is an interesting branch of science but I don’t believe it is particularly dead on accurate.