HTF III, an example is probably the best way to show you how the chart is used.
Let's say you've made a satisfactory exposure at a particular enlarger height and want to now make a larger print from the same negative. Before you increase the enlarge height, measure the current easel-to-lens (lensboard is close enough) distance. This is read on the horizontal scale. Now raise the head as desired for the larger print and again measure this distance, which is read on the vertical scale. Then find where the two readings intersect in the family of curves and note which of the 'ray-lines' is closest to that intersection. That 'ray-line' tells you how many stops (or fraction of a stop) to increase the original exposure time. The 'ray-lines' range from 0 (no change) to 2 (two stops), with intermediate changes shown in increments of 1/6-stop.
This will get you close enough to the new exposure, but you'll likely want to do another test strip anyway at the higher head setting to fine-tune this result. As stated on the chart, paper reciprocity can affect this new exposure.