Basically, the above answer is a good summary. I will add a few observations to it. The poem is a type of fable, a tale that uses animals to develop a moral lesson. You can tell by the examples of alliteration, such as "Bingle Bog" and the rhyming nature of the poem that it is intended to move swiftly and to draw in children.
However, the moral is one for all ages. The frog is the evil villain. He is rude, manipulative, greedy and without emotion. The nightingale is the innocent and naive victim. Villains hate beauty, and the frog is no different. He hates the beauty of her singing and is jealous of the attention her singing brings to the bog. He decides to capitalize on the situation by pretending to befriend the nightingale, who has never been subjected to this kind of manipulation. He charges admission and continually scolds the poor bird, pushing her until she dies.
Notice the animals have no names. The nightingale can represent anyone with a true talent and love for something. The frog is the person who uses this talent for his own means. Think of a parent whose child is gifted in a sport. How many times can that parent scold and push that child to hate a sport he once loved?
This is the moral of the poem