The argument is that a ward system, which requires representatives to be elected from districts within a city in the same way that states elect congressmen from districts within the state, creates a system where councilmen are more interested in advancing the interests of their ward than advancing the interests of the city as a whole. The Dispatch argues that Columbus has been successful compared to other midwestern cities and at the same time has more buy-in and confidence from its citizens, as evidenced by the willingness of voters to approve a 0.5% city income tax increase in 2009.
The discussion warrants a comparison to the US legislative system. Both of the federal government's two houses are elected on a "district" system: The US Senate elects by state and the House of Representatives by districts within states.
Congress is notorious for its low approval rating amongst Americans. Current data places that approval rating at about 17.5%. On the other hand, it is difficult to find any of the 535 senators or house members who have an approval rating so low. One explanation for this is that each member has reason to cater to and promote his or her own state or district, but has little electoral incentive to advocate for the good of the country as a whole.
Contrast this with the European system of parliamentary voting. While there are many variations of the system, the basics come down to voting for a party rather than a person. In this system, the entire country will cast votes for parties, then each party will get a portion of the legislature's seats equal to the proportion of the total votes they received. Theoretically, these legislatures would then be more interested in advancing the country as a whole. This would mean that their approval ratings amongst their citizens would be higher than that of a federal system such as that of the United States.
This seems to bear itself out in the polls, at least in one example. A 2011 story about the British parliament measured its popularity spread (% approve minus % disapprove) at just over -30. This is not good by any means, but looks like a pittance compared to congress's current spread of -58.
Part of the reason the US system is built in such a manner is because it was conceived in an era of strong states' rights. Since the founding, the power of states has deteriorated greatly, but the system built on their prevalence persists.
The advantage of the ward/district system is that it allows for every geographic area to have an advocate. While an at-large/European system might be more willing to sacrifice a certain geographic area's good for the good of the whole, this becomes harder in the ward/district system. And in some cases, the good of a geographic minority should trump the greater good. The tyranny of the majority raised its ugly head when the state of Ohio voted to place a casino in Franklin County against the will of the residents of the county.
While an at-large/European system is advantageous to advancing a more holistic good, it does sometimes make the mistake of overlooking the needs of geographic minorities.