The inductor and capacitor form a resonant circuit, which will pass only a specific frequency - the one you are tuning the radio to recieve. You normally tune it by making either the inductor or capacitor adjustable.
edit: As described in How does radio receives signal from particular station? it's very much like a pendulum.
Current flows freely in the circuit only at one particular rate - set by the inductance and capacitance. Try to make the circuit oscillate at any other frequency will just cause the energy to be absorbed by the capacitor and inductor and so only changing electrical signals at the exact right rate are passed.
The maths are a bit complicated but there is a good explanation here.
It is surprising that you can pull a particular signal out of all the random frequencies arriving and being detected by the antennae - that was the point of my piano+morse code example in the other question.