"The flame on its tail indicates Charmander's life force. If it is healthy, the flame burns brightly."
Right from the SoulSilver/HeartGold Poke'dex entry. Previous entries never outrightly state "death", but it's heavily implied in the show that if a Charmander's flame goes out, it will die - because it is a measure of it's health.
I think in Charmander's introduction episode in the first season, it was raining... it could have gotten sick, which was why the flame was getting less stable.
It could have also been symbolic of hope in that episode, since Charmander's beautiful soul of a Trainer had abandoned it.