The story of the bat

Coat of arms of the city of Valencia.

Valencia and the Balearic Islands were conquered by King James I of Aragon during the first half of the 13th century. After the conquest the king gave them the status of independent kingdoms of whom he was also the king (but they were independent of Aragonese laws and institutions). The arms of Valencia show those of James I, King of Aragon.
The unique crowned letters L besides the shield were granted by King James. The reason for the letters was that the city had been loyal twice to the King, hence twice a letter L and a crown for the king.
There are several possible explanations for the bat; one is that bats are simply quite common in the area. The second theory is that on October 9, 1238, when James I was about to enter the city, re-conquering it from the Moors, one bat landed on the top of his flag, and he interpreted it as a good sign. As he conquered the city, the bat was added to the arms.