Transformers consist of the primary and secondary coils, and the iron core they are coiled around.
When AC current flows in the primary coil, and assuming the secondary coil is in a complete circuit:
[ul]
[li]The changing current of the primary coil makes a changing electric field[/li]
[li]The changing electric field induces a changing magnetic flux mostly within the iron core (since it has a much higher permeability, that is, it's easier to generate a magnetic field in the iron core)[/li]
[li]The changing magnetic flux generates an electric field around the iron core, this electric field induces current within the coils of the secondary by accelerating the electrons[/li]
[li]Like this, most of the energy from the primary coil is transferred to the secondary coil[/li]
[/ul]
[link href=https://web.archive.org/web/20180827084639/http://www.physics.usyd.edu.au/~khachan/PTF/Transformer%20Explanation.pdf]How does the primary coil 'know' when something is connected to the secondary coil and transfer energy[/link]
[link href=https://en.wikipedia.org/wiki/Magnetic_energy]Energy can be stored in the magnetic field[/link]
Remember:
$$\phi=BA\ and\ F=qvB$$
