For superconducting circuits coupled to a resonator, we get a shift in the transmission spectrum (vacuum Rabi splitting) because of the interaction of the qubit and the resonator.
Then one goes to the dispersive limit where we have large detuning $\Delta$. Then you can approximate the Jaynes-Cummings Hamiltonian and see the interaction term as perturbation. Afterwards the Hamiltonian shows a cavity frequency shift of $$\pm g^2/\Delta \, .$$ With this we can determine the state of the qubit.
Why do we have to detune the system when we already see a splitting because of the interaction of the resonator and the qubit? Can we perform non-demolition measurement only in the detuned case? The splitting for the detuned case is really small, when $\Delta$ is big but is it still bigger than for the not detuned case?