Op-amps have three basic properties: near infinite input impedance, near infinite open loop gain, and a virtual short between the two input terminals (when used with negative feedback). The short causes Vref to appear at the inverting (-) terminal of each op-amp. As a result, no current is flowing through Rc. However, when a switch is enabled, each of the sensor array resistors in that row can now conduct current. Where does this current come from? The op-amp's output through the feedback path. The final output from the analog switch comes from the current being drawn due to the sensor multiplied by Rg in the feedback path of the op-amp. In an equation:
V(read) = Rg * Vref/R(sensor)