Never ask me for references, I read random articles and infer the rest...
The feedline interacts with the nearby EM field, certainly -- but because the wires are close together, they experience the same induced voltage/current. The transmitter connects through a balun, rejecting in-phase voltage/current. The feedline is also perpendicular, so its polarization is opposite that of the dipole element.
This is key: the field only sees the net total current or voltage of the wires in a small region. The field is large and poofy -- on the order of a wavelength -- while the wires are closely spaced.
That's why we can make the distinction between the field "inside" a transmission line, the differential mode; and the field "outside" it, the common mode.
The folded dipole is a differential to common mode converter, arranged in such a way that it radiates as a dipole antenna.
Remember that electric field, or magnetic field, doesn't simply
be; work is required to generate a field. This work corresponds to the current flowing through, and the voltage present on, the elements.
While there is no physical connection to the tips of the dipole elements, current nonetheless flows into them. This current is called the displacement current. The current is drawn by all points along the element, so the current in the wire is also not constant along the element.
Likewise, the magnetic field generated by that current, is concentrated towards the center (near the feedline).
It just so happens that, the fields are large enough, correctly spaced, and correctly phased (because remember, everything is moving at the speed of light, as well as oscillating at the radio frequency), to couple the electrical signals into free (propagating) EM waves in a certain direction (namely, the radiation pattern of the antenna).
Tim