If someone told you only "a capacitor passes AC, and an inductor blocks AC", then it's no wonder that you are confused. The statement is too simplistic. It omits critical variables such as the values of the components, and the frequency of the AC.
A resonant system requires math to explain exactly what's happening, but I'll try some non-mathematical hand-waving.
Capacitors and inductors are energy storage devices. A capacitor stores energy in an electric field. An inductor stores energy in a magnetic field. In an LC resonant circuit, the energy transfers back and forth between the capacitor and the inductor.
Maybe a "water circuit" analogy will help you. Imagine a loop of pipe filled with water (the circuit). Water flow is equivalent to electric current. Water pressure is equivalent to voltage. Pretend that the water has insignificant inertia. Now insert an elastic diaphragm (capacitor) into the pipe. Water can't flow through the diaphragm, but a pressure difference stretches the diaphragm, storing energy in the elastic. Now also insert a flywheel impeller (inductor) into the pipe. Water can't flow through the impeller until a pressure difference gradually causes it to spin, storing energy in its rotating mass.
Now apply a momentary external force to the water in the circuit. Notice how the water, diaphragm, and impeller begin oscillating back and forth. The pulse of energy that you applied is now transferring back and forth between the diaphragm and the impeller. The frequency of oscillation depends on the values of those two devices.
A real system will have some friction (resistance), so the energy will gradually dissipate as heat, gradually decreasing the amplitude of the oscillation. If you apply a continuous series of external pulses at the circuit's resonant frequency, you can maintain continuous oscillation. If you apply the pulses at the wrong frequency, the oscillation will tend to stall. Same thing happens in an LC resonant circuit.