Inductive Charging

Faraday’s law: a changing magnetic field will produce electromotive force (i.e., voltage) across a conductor.

Oersted’s law : an electric current creates a magnetic field. 

RF (Radio Frequency) Charging

Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. Radio waves have frequencies as high as 300 GHz to as low as 3 kHz. Like all other electromagnetic waves, they travel at the speed of light.

Does electromagnetic radiation induce current on a conductor when it interacts with the conductor? What actually happens in the microscopic level?

Yes it does and this how a wire antenna delivers currents and voltage to a receiver. Electromagnetic fields have both electric and magnetic fields components. These fields oscillate thus they have both positive and negative half-cycles. The electric fields remain outside the metal but they attract electric charges (electrons) to the surface of the metal - or repel these electrons when the field is negative. The magnetic fields induce currents in the metal's surface which are moving along the surface in direction perpendicular to the magnetic field. These charges and currents can be be calculated precisely using Maxwell's equations.

At high frequency there is a skin effect that doesn't allow fields to penetrate deep inside the conductor other than a thin skin, which gets thinner with higher frequency. For example, at 1 MHz the skin depth of copper is about 0.1mm although the wavelength is about 300m. This means that at high frequencies the fields remain outside the conductor and the conductor is just guiding the adjacent waves.

Here is a simple circuit that can be used to collect the energy in radio waves and convert the current into a DC:

For more details on above circuit, refer to https://m.youtube.com/watch?v=XpLCK88nVgU