This answer is largely conjecture on my part, and doubtless others have different viewpoints. I do agree with the answer given that he is a character added for the enjoyment of children, and people hate him for the same reasons they hated the other kids characters.
But here are my thoughts.
Jar-Jar is a fool. A buffoon. He is a clumsy, stupid coward, and worse, he is from a race of soldiers and warriors. No wonder they banished him. He's lucky they let him live. His only saving grace is that he is not malicious in any way. You could be buddies with him. Though, granted, he is that buddy you keep away from anything breakable or sharp.
During Episode 1, the gungans have very little power on Naboo. They keep to their underwater cities and secret hideouts. But at the end, a new friendship is forged between humans and gungans.
By the time we reach episode 2, we can see that they have quite a bit more political clout. Naboo has two senators. One human, one gungan. Why Jar-jar, then? Why not Boss Nass? He's certainly a powerful and popular leader among the gungans. My guess is, Padme pushed for him. She is not a stupid woman, and is a pretty clever politician. She knew that Jar-Jar had become popular after being made a General at the Battle of Naboo. She also knew him for the fool he was. He had no knowledge of galactic politics, and was pathetically eager to please. With him as senator, Padme would effectively have two votes. This sounds a bit manipulative for Padme as we know her, but she was Queen and later Senator. Her goals were always noble, but she knew how to play the political game. (According to Wookieepedia, it was Palpatine who made him Senator, but I don't see Padme objecting. It was from a comic, anyway. Very Extended Universe.)
By the time we get to the clone wars, Palpatine uses Jar-Jar to make a speech asking to give Palpatine emergency powers. Look at Jar-Jar in that speech. He has no idea what he is doing. He's parroting a speech Palpatine doubtless wrote, then when the Senate erupts in applause, he stands there with a big dumb happy smile, as if thinking, "Theysa likes me! Mesa big bombad Senator! Mesa getting applause!"
His role is to be a fool. A harmless fool, at first. A lucky fool. And finally a fool that is used by more intelligent men to help bring down the government.
Just my thoughts.