There are many different versions. In another, a priest tries to catch the tanuki to have it for dinner, but the tanuki tries to escape by turning into a tea kettle. Many Japanese children have never seen a traditional tea kettle, but non-Japanese children have probably never seen a tanuki either.
Japanese people think of tanukis as having big, round bellies sticking out in front.
It just so happens that Japanese tea kettles also have rounded middles. Well, anyway, lots of people thought that tanukis and tea kettles were similar enough to make a good story. This story was so popular because, unlike children today, Japanese children in the Edo Period still used tea kettles in their everyday lives. The kind of tea kettle we are talking about here doesn't actually hold tea, but instead is used for boiling water for the Japanese Tea Ceremony. This taste contest, in which the drinker tried to guess the origin of each variety of tea, was one example of the new trend of drinking tea for fun.