Radio waves are produced when the electric field rapidly changes: there has to be an alternating current.
An electric field spreads out into space. When you change an electric field, the distant parts of it do not change instantly. The change is limited by the speed of light. If you fluctuate the electric field, you therefore create a wave.
You can think of it as space being permeated everywhere by an electric field; your circuit just creates a disturbance in it, like disturbing the surface of water. The disturbance travels away at the speed of light, like ripples in a pond. If your circuit just has steady DC flowing through it, the disturbance occurs just when you switch it on and when you switch it off.
(Indeed, electric equipment causes interference when it turns on and off: relays, switches, the commutation of electric motor brushes, or anything that generates sparks: all radiate and can interfere with radio communication, or with sensitive equipment.)
Radio-transmitting circuits are optimized for radiating; they deliberately do things that designers try avoid in circuits that must minimize their radiation (which is most circuits). Transmitters amplify some high frequency AC, and energize an antenna.
There are many kinds of antennas and how they all work is a big topic. One example of an antenna is simply a dipole of half a wavelength: two long conductors pointing in opposite directions, each a quarter wavelength long.
