Why do old transformers make sounds, but new ones don't? What I think is that maybe insulation is gradually removed between plates, and eddy currents start passing from one plate to another and this produces sound. Am I right?
 A: Transformers generate oscillating magnetic fields at the mains frequency and the fields produce an oscillating force on:


*

*anything nearby that's ferromagnetic (like the core)

*anything nearby that is carrying a current (like the windings)
The sound you hear is because various bits of the transformers are moving in response to the oscillating fields and this movement creates the sound. It is a purely mechanical process.
When transformers are new they are very tightly constructed so nothing within them can move in response to these oscillating forces, and they generate no or minial sound. However over time the transformer loosens up and this allows some movement and the corresponding vibration. I would guess the major cause of this is temperature cycling and the associated expansion and contraction.
A: Cores of the transformers a made of ferromagnetic material, which can change the shape due to magnetostriction phenomena. When transformer works with typical grid frequency of 50Hz..60Hz, it can be heard. As it was answered above, generated noise depends on mechanical construction. New power supplies use frequency converters, which drive the transformers with frequency high above audible 20kHz, so they are silent (when are stable). 
