There is, of course, a prevent-graft-and-corruption angle to this, but the primary arguments are cost and participation. Publicly financed elections will limit the huge campaign advantage --through fundraising--that incumbents enjoy. It will also increase the pool of qualified people who might consider running, no longer limiting it just to people who can afford to take six months off to raise money. It will end the massive transfer of wealth to local TV stations, which is where much of the campaign money ends up going. And finally, just as a thought experiment, run your eye over the Portland cityscape and ask yourself how many expensive projects the City would not have undertaken if elected candidates weren't beholden to big donors.
And, of course, although public financing was implemented, to the best of my knowledge the world hasn't ended.