Again we point to the properties of Network Effects. Like eBay is a network, so too Visa is a network. In a "two-sided" market, more buyers equals more sellers. The goal of Interchange fees is to optimize both sides.
For example, no barriers for buyers on eBay, brings more buyers to the marketplace, thus attracting more sellers to the value of both.
Payment card systems are not the only case such two-sided markets exist. Software industry, video games, internet portals, medias and shopping malls. In all these industries, the crucial challenge for the platform is to get both sides of the market on board while making a profit overall.
The economic value of Adobe PDF software comes in great part from the fact that any potential reader can download for free the complementary software. Software programs like Adobe distribute readers from free and charge the writers and content providers. This price structure results from the fact that readers have a lower willingness to pay for the software than the writers.
Thus Interchange fees are crucial for payment networks to find an effective method for balancing the prices on the two sides of the market.