What does this graph explain?
A graph is generally meant to be interpreted as y being a response to a change in x. This graph highlights that as the partial pressure of oxygen $P_{O2}$ increases, so does the percentage of hemoglobin that are saturated in oxygen. The relationship is not linear (unlike you seem to suggest) but is logistic.

The graph also highlights how three different level of affinity would affect the relationship between $P_{O2}$ and the fraction of hemoglobin that is saturated and highlights side consequences of such affinity variation.

Finally, with vertical dotted lines, the graph indicates what the $P_{O2}$ is in the lungs and in the other tissues.


  why a decreased affinity is a shift to the right and an increased affinity is a shift to the left?


Consider a single point in the x axis such as $P_{O2} = 40$ for example. If affinity is increased you would expect that the percent saturation of Hb would increase, right? This causes a shift of the left.