Jay Gatsby in The Great Gatsby has carefully crafted his persona, apparently, before Nick ever meets him in the narration we read. He is purposely mysterious, partly, as far as we can tell, because of his business, which he, we assume, wants to keep secret. It seems important for him to keep a low profile, at least as far as business is concerned.
His mystery also comes from the fact that in some circumstances he is reticent and shy. He is not comfortable in some social settings. Even at his well-attended and lavish parties, he barely talks to anyone. He seems to live very much within his own mind.
Part of his living within himself is also due to his obsession with Daisy. If you are not Daisy or somehow connected to Daisy, he really doesn't have any natural interest in talking to you.
Finally, his mystery is designed to captivate, specifically to captivate Daisy. Gatsby has an aura about him. He seems bigger than life (he is referred to as "great" in the title, though this is partially ironic), and he has to be, he thinks, to recapture Daisy.
Gatsby's own mystery, then, lends itself to rumors. You put a bigger-than-life figure in a mansion frequented by wealthy guests, and you will get rumors. Especially when he keeps his business affairs secret and his thoughts to himself.
The author allows for and creates these rumors to characterize Gatsby and give credence to the type of person that might have been a regular fixture in 1920s America.
The rumors listed above served to make us second guess Gatsby's identity. Much illegal and/or immoral activity was going on in the 20s and wrapping his character in what might have surrounded a character who two lives like Gatsby seems to during Prohibition.
Tom professes later that Gatsby was likely a bootlegger. This could relate to someone like Al Capone. If Gatsby was involved in shady dealings, he likely lived the high life in front of his friends but did dirty deals behind the scenes.
There are a number of rumors about Gatsby, some true, some not true. There are rumors that say that he has killed a man. We do not know if those are true or not.
There are rumors that say that he was related to the ruler of Germany (from World War I) -- the Kaiser Wilhelm. These are clearly not true.
There are also rumors that say he has made his money illegally, through bootlegging and such. These seem to be true.
To me, the rumors are meant to show how aloof and mysterious Gatsby is to the people who come to his parties. They do not really know him even though they are at his home so often.