Why dc component I must flow entirely through the load resistance R in buck converter?
The key to the question is that they asked about the "DC component".  Capacitors and inductors influence the AC properties of a circuit.  DC implies steady-state, as in no time-varying components.  You can usually think of a circuit as steady-state by replacing inductors with shorts (or low value resistors) and caps as open circuits.  In that case you would have I passing simply through R.

Yes, the capacitor charges initially when switching, but that current is not steady-state, it's time-varying, so it's not part of the DC component.  The AC component, as the question goes on to explain, divides between the cap and the resistor.

Think of the AC and DC components being added together to provide the final result (called superposition).  In the image below you see AC voltage ripple added to DC voltage.  This is for voltage but it would be the same principle for current.