How do radio waves physically "travel"? If sound waves physically travel by one molecule bumping into another and the chain reaction eventually reaching my ear, how is a radio wave emitted and how does it "move"?
 A: They are self propagating. An oscillation in the electric field results in an induced one in the magnetic field, and vice versa repeatedly. Thus, it propagates itself through space with these repeated inductions regardless of physical surroundings.
See http://www.physics.usyd.edu.au/teach_res/hsp/u7/t7_emr.htm.
A: A radio wave is an electromagnetic wave ( EM wave ). EM waves are of different categories depending upon their wavelengths and the one with the longest wavelength is the radio waves.   Unlike sound waves which is a mechanical wave which require an elastic medium to travel, electromagnetic waves consist of  synchronized oscillations of electric and magnetic fields. The electric field and magnetic field oscillates perpendicular to each other and the direction of propagation of the  EM wave is perpendicular to both the direction of oscillation of the magnetic and electric field. 

The energy of the wave is transferred as oscillating electric and magnetic fields.

Hence, radio-waves doesn't require a physical medium to travel.
A charged particle at rest only produces an electric field. A charged particle at uniform motion will produce a magnetic field. An accelerating charged particle ( acceleration as a part of oscillatory motion ) produces an EM wave ( with the magnetic and electric field oscillating perpendicular to each other as shown above. ). i.e.,to say more profoundly, accelerating charges produce changing electric and magnetic fields.  Changing electric fields produce magnetic fields and changing magnetic fields produce electric fields.  This interplay between induced electric and magnetic fields leads to propagating electromagnetic waves.
You may note that any accelerating charges can create ripples of oscillating magnetic and electric fields. But, to get a well defined EM wave, an oscillatory motion related acceleration is required.
This would be of more help : https://en.wikipedia.org/wiki/Electromagnetic_radiation
