Electromagnetic waves
Also see... Why is an inductor not a good antenna?

Simple answer:  During any positive half-cycle of alternating current, a transformer coil sends out a magnetic field which surrounds the coil.  Next, the AC wave decreases to zero, and the magnetic field then collapses inwards. (The energy in the magnetic field is recovered by the coil.)   In other words, the magnetic field "balloons outwards" but then gets "sucked back in."   That's how inductors work, for AC.

Loop antennas are slightly different.   They do behave the same as inductors, with a magnetic field which balloons outwards and then gets pulled back in.  But with loop antennas, the speed of changing fields is too fast, and the field-changes can only move at the speed of light.  With loop antennas, only part of the field gets pulled inwards and recovered by the coil.   Another part flies outwards and is lost.

So, an inductor or transformer recovers 100% of the magnetic field whenever the current falls to zero  ...but a loop antenna only recovers a smaller percent of the field, while the rest gets radiated as waves.

Here's an simulation of an inductor being driven with AC, from an EM physics course at MIT.  The inductor in the video is microscopic, and located in the center of the screen.  Its magnetic field expands and contracts, with all the magnetic energy being recovered during each inwards cycle:

vid: MIT teal project, nearfield AC field



Above is the inductor, while here below is their simulation of a loop antenna.   Only part of the field gets drawn back into the inductor.  Some of it escapes and flies away:

vid: Antenna w/nearfield

Finally, here's the field-pattern for a pure antenna.  The "nearfield" zone with it's expanding/contracting pattern, that part is tiny, and down in the center.  The surrounding "far field" shows the radiated EM waves.

vid: Antenna far-field view

MIT 8.02 course also has a simulation of an inductor where the current suddenly changes to a higher value.  The magnetic field pattern enlargers, and also a transient-wave of pulsed EM gets launched:

vid: Inductor w/step-current

What's the difference between a loop antenna and an inductor?  A loop antenna is physically large compared to the lightspeed delay, and to the AC wavelength.  When its usual fields start acting like waves moving across the antenna, then it loses the ability to "pull in" the whole field-pattern.  Inductors are different; they're microscopic when compared to the EM waves.  That means their fields can all change together at the same time, with no waves involved.  They can "pull in" the entire field as a whole.

Of course all inductors leak a little bit.  For this reason even the smallest inductor can become a transmitting antenna.  Just give it a million times higher current than would be needed by an electrically-large loop antenna having a quarter-wave diameter.

Reference: MIT video animations for Physics 8.02 E&amp;M