The actual reason why Charizard began to listen to AshstyLIShT In the original episode where Ash caught Charmander he found it on a rock because it was abandoned by it's original trainer. When Ash, Brock, and Misty visited the Pokemon Center they overheard a bunch of other trainers talking and realized one of the trainers amongst them named Damien was the one who abandoned Charmander.
One of his friends asked him about the Charmander and Damien responds saying "It was so weak that it couldn't even beat the weakest opponent". Upon hearing this Brock responds to what he said with "Charmander might be weak against water Pokemon, but if its trainer works hard it could become strong". This response makes no sense within the context of what they were saying because there was no mention of Water Pokemon, however this is due to a dubbing error. In the original Japanese version Damien says "Charmander was so weak, it lost against a Poliwag", to this Brock responds saying "Charmander is weak against the water type, but a good trainer can help it overcome this".
Then at the end of the episode Damien admits to Charmander that it abandoned it for being to weak but now that it has toughened up he'll take it back, he also admits that he has no interest in raising his Pokemon and just wants to use them to win battles.
Now if we look at the episode where Charizard starts to obey Ash it starts off with it battling against Poliwags final evolution Poliwrath. However it didn't listen to anything Ash said and tried battling the Poliwrath on its own which ended with it losing once again because of its type disadvantage. However this time its trainer didn't abandon it for losing and even promised to become a better trainer so they can become stronger together. At this point Charizard starts having flashbacks from when it was abandoned the first time it lost and realizes its new trainer would never do anything like that because he cares about it.
At the end of the episode Ash ends up having a rematch against the trainer with the Poliwrath however this time Charizard and Ash work together in order to overcome the type disadvantage and defeat the Poliwrath.
Also it is worth noting that in the Japanese version during the battle where Charizard defeated Poliwrath it played a theme called "type wild" which is an uplifting theme. The only other time this theme was used during a battle was when Ash's Infernape defeated Pauls Electivire which was another case of an abandoned Pokemon overcoming something with Ash's help.
Holy crap... - TwilightKitsune