Hi,
The Barkhausen Effect is something different but it may help here.
A magnetic material is made up of structures called magnetic domains, and they act like tiny somewhat independent magnets. When a current is applied that sets up a field in a given direction, and if that direction is the right direction it causes a domain to 'flip'. Flipping of a domain is pictured by thinking of a tiny bar magnet that is allowed to rotate about an axis. It tries to align itself with the field and when the field reaches a certain level it flips or at least aligns itself with the field. Depending on how much it moves it creates a small current of it's own because it's a magnet who's own field suddenly contributes to the total field (and that's the generator effect) and this is called the Barkhausen Effect here. This helps to understand the domains.
But a given bulk of material only has so much volume, and a limited volume has a limited number of domains. Once all of the domains are aligned, there are none left to flip so the magnetic material becomes much less magnetically active. When a material is magnetically active it means that it can react to a magnetic field set up by say an electric current. When it becomes inactive it no longer reacts to that electric current. When it is active it limits the current (over time) but when it is inactive we only have the air to act as the magnetic medium and air has 1000 times less capacity (or even less) to react to a magnetic field than a true magnetic material. This means once the magnetic metal becomes totally 'used' (all the domains flipped) the only thing left is the air and that doesnt limit the current (over time) as well as the available domains did. That means once this happens the current usually rises and it rises very quickly.
While this is taking place the AC current is causing an interaction between the laminations of the core. Usually these low frequency cores are made up of thin plates of metal which react well to a magnetic field, and they are placed side by side to increase the total bulk of the material and thus elevate it's usefulness as a magnetic core with additional domains. Unfortunately because they are physically different pieces of material, they are attracted to and repelled from each other during the AC cycle, and this causes them to vibrate and bang into each other. Since the higher the current the stronger they attract and repel, the harder they bang into each other and so the audible noise increases. Now normally the increase in current is not enough to cause significant noise in the core or the coil, but as it increases above the design level it starts to make a lot more noise. This is especially true when the plates have not been vacuum varnished because the varnishing partly glues them together and limits the movement somewhat. You dont want to be in the same room as a 10 Killowatt 60 Hz transformer that has not been varnished even if it is not saturating
To explain saturation from the standpoint of inductance, a given coil and core has a given inductance until it partly saturates (some domains flipped) and then the inductance falls to a lower value. If it saturates deeply (most domains flipped) it means that the current goes up in the coil and it could go up very high until the drive element (like a transistor) blows out.
If we look at the equation for an inductor:
V=di/dt*L
and rearrange it a little to solve for di:
di=V*dt/L
we see that if the inductance (L) gets smaller the change in current (di) gets larger. That's how saturation affects the circuit. the inductance falls, the current rises. This equation is linear in L so at first it looks like halving the inductance L will only cause an increase in current of two times, but unfortunately the curve of the magnetic material is very nonlinear such that as the current rises the inductance falls faster and faster, so the current ends up looking like a sharp spike that can rise up much higher than it was originally designed for. That happens because the permeability of the core drops off much quicker and that means the inductance falls fast.