Reading the Wiki page doesn't really explain why it has three legs or the significance therein. It just states a bunch of history regarding the appearance of such a creature in Chinese and Japanese imagery going back to the Neolithic age. Is this because we don't know or am I just missing something?
The number 3 has various meanings within Chinese numerology:
While there is no consensus amongst scholars as to the origin of this myth, eastern numerology helps us to glean its meaning. The three-legged crow is often a messenger, as Yatagarasu was to Amaterasu, the sun goddess of Japanese mythology. With 3 representing multiple stages of life and being, it is able to commune with both the divine and the mortal. In Chinese legends, the three legged crow Jīnwū often travels from the heavens to the earth, again reinforcing this theme of transcendence between worlds.
This article expands on the sun motif, as well as offers possible explanations of how the three-legged crow's motif and legends were influenced by various cultures.