It's not about obesity, but the doubt and self-hate gays feel when told by society that they're an abomination.
Charlie never thought he was a blight on society because of his sexual orientation, but after being told that for so long, he began to question his own goodness. The movie was about him in his last days searching for any reason, through his daughter, why the world's view of him might be wrong. When he dies at the end, it's suggested that he went to heaven, meaning God doesn't share the view that many in the world do towards gays. It's a very affirming, pro-tolerance/acceptance message.
The obesity is a metaphor for the ugliness and revulsion that homophobic people see/feel when they look at gays. The use of obesity in this way is problematic though, because the movie leverages fat-phobia to push gay-acceptance.