How exactly are radio waves produced from a current in a circuit itself?
Don't worry about photons unless you want to venture into quantum physics. A photon is the quantum of electromagnetic radiation, which is also a wave. I've yet to find an application in RF engineering where quantum effects are relevant.

In all electronic circuits, there are two fields: an electric and a magnetic. The electric field is associated with voltages, and the magnetic with currents.

We have components that make strong electric fields: capacitors.

We also have components that make strong magnetic fields: inductors.

In each of these components, we think of one kind of field as dominant. But consider what happens if we rapidly change the magnetic field through an inductor, say by passing a strong permanent magnet through it: a voltage will exist between the terminals of the inductor. This voltage is an electric field. We call this Faraday's law of induction.

A similar thing can happen to a capacitor. To change the electric field, there must be a current. Or if you manage to change the electric field, you will find a current somewhere. Manipulating the electric field inside a capacitor is rather more difficult than dropping a magnet through a coil, but if you can construct an appropriate experimental apparatus, you will find this is true.

Thus, a changing electric field can create a magnetic field. A changing magnetic field can create an electric field.

Electromagnetic radiation is these two fields creating each other in free space. The electric field changes, creating a change in the magnetic field just in front of it, creating a change in the electric field just in front...



To get these fields to radiate away in free space like this, you must create both, in phase, perpendicular to each other. This is why a capacitor is not a good antenna: it creates a strong electric field, but the magnetic field is relatively small. It radiates a little bit, but mostly the energy is stuck in the electric field, unable to radiate away because it has no magnetic field to carry it away from the capacitor. Same is true of an inductor, with current and voltage, magnetic and electric exchanged. See Why is an inductor not a good antenna?

Antennas are just leaky inductors or capacitors. Many antennas are equally both at the same time, such that their impedance is purely resistive at the design frequency, rather than inductive or capacitive. Through clever geometry, they create magnetic and electric fields perpendicular and in-phase, which then radiate away.