Most of the corruption during Harding's administration stemmed from cronyism - Harding took most of his 'friends' (business acquaintances, mostly, and political bedfellows) to Washington and gave them plum jobs, like Cabinet members and such. Albert B. Fall, former senator from New Mexico, was the worst offender, first gaining control of the Teapot Dome oil reserves from the Navy, and then leasing them to oil interests for a bribe. The Attorney General of the time, Harry Daugherty (crony from Ohio), was also indicted for his extreme reluctance to prosecute anyone in the Administration. It appears that Harding was not connected to any scandal, but the fact that he appointed these individuals to positions of power speaks volumes about him.
Also, Harding first gained popularity in his native Ohio as the editor for the Marion Star, and parlayed that into successful bids for State Senator, Lieutenant Governor, and later U.S. Senator. His newspaper roots often showed through in his self-written speeches - he'd almost assault the listener with a near-hypnotizing deluge of words. This is not to say that the speeches were good, by any means. Often, they said nearly nothing. But it's so hard to dig through the phrases that most people just assumed that Harding knew what he was talking about. Two examples:
"Progression is not proclamation or palaver. It is not pretense nor play on prejudice. It is not of personal pronouns, nor perennial pronouncement. It is not the perturbation of a people passion-wrought, not a promise proposed..."
"America's present need is not heroics, but healing; not nostrums, but normalcy; not revolution, but restoration; not agitation, but adjustment; not surgery, but serenity; not the dramatic, but the dispassionate; not experiment, but equipoise; not submergence in internationality, but sustainment in triumphant nationality...."
These speeches, combined with his look (Daugherty, explaining why Harding was pushed for a Republican nomination: 'He looked like a president.') helped with his initial success.