Flater is correct in his assessment that the statues are representations of the previous avatars and that them lighting up is more thematic rather than functional.
However, I would like to address the specifics of the question: each of these statues are representations of the previous incarnations of the Avatar. The ones shown outside of the Southern Air Temple are located in other temples scattered around the world, there are at-least one of these temples located in each nation.
It's never explicitly explained how these statues are created, but it is logical to assume the friends and followers (the order of the white lotus) of the latest Avatar are likely the ones who construct these statues.
As for the statues being damaged having any effect on the current Avatar, there is evidence to the contrary. In a later episode of season 1 (episode 7 and 8):
Aang travels to the fire nation to commune with Roku, the previous Avatar, at his temple. At the end of the episode Aang, while channeling Roku, awakens the volcano Roku's temple sits upon, forcing everyone to flee as the temple and the statue inside of burn and melt into the flowing lava. The destruction of this temple and its statue has no lasting effect on Aang.