Prerequisites: calculus, derivatives; E&M, static fields; definition of capacitance.
Consider the construction of a capacitor:
We have two parallel plates (or anything equivalent to them: a roll, a stack of plates, big sheets folded over..), between which a dielectric is placed.
When we apply voltage, we're setting the electric field in the dielectric. How much charge develops in return, depends on the dielectric constant, and the general response of the material.
We don't need the particular geometry of a component, we can resolve this question by considering material properties alone. Any given component will be proportional to these properties, rescaled by capacitance (area) and field strength (layer thickness).
If you've read about magnetism, you might've seen the B-H curve. Ferroelectric materials are the same way, but for electric fields. Consider the P-E curve of such a material:
From: https://resources.pcb.cadence.com/blog/2020-understanding-a-ferroelectric-hysteresis-loop-in-electronics
This says that, as field strength E increases, at first the polarization charge P goes up quickly (the dielectric constant dP/dE / ε = κ is high), but gradually it tapers off, and the material is saturated. That is, charge rises only slowly as field strength continues into saturation, asymptotically approaching κ → 1.
For BaTiO3 materials, ε might be reduced by 50% for field strengths in the 1V/μm ballpark, with breakdown occurring in the 100V/μm range, depending widely on dielectric quality of course. Typical high-density capacitors (say 10uF 6.3V in 0805) might have layers of a few 100 nm thick, so the reduction with bias can be strong even for small voltages.
But something is missing: the AC capacitance doesn't change; I mean, the average value will fall as peak amplitude goes into saturation, yes, but there's no shrinkage at small amplitudes. What gives?
The trick is, they lie to you. These B-H curves are rarely plotted in detail -- it's a smoothed-over cartoon of the real material, or at least a lie of omission. As it happens, if we reduce the amplitude and measure the B-H curve of a small loop, we find its slope shrinks. We have a diagram more like this:
It's tilted in the middle. There exists not just one, but a family of curves where, for large signals (going deep into saturation), the curve just goes far enough fast enough that you don't see the pinch; for modest levels, they are mostly linear (avoiding the saturation peaks); but for smaller curves, closer to zero, they slope down. In ferromagnetic materials, this gives the initial permeability, and the analogous effect appears in ferroelectric materials.
So, what the AC bias is doing, is pushing the curve out of the sloped initial region, into the bulk of the curve space where it's more linear, and (for the finished capacitor) closer to nominal value.
DC bias serves a similar purpose: notice the derivative at Vdc = 0 is flat, or maybe even a little positive. Sometimes it is positive, and the curve rises to a peak a few percent above nominal value. DC bias pushes the curve out of the initial region, at least in part.
When we measure capacitance, we're usually looking at cycle-averaged parameters: the in-phase and cross-phase components at the driven (sine wave) frequency. Thus, ignoring harmonic distortion, and reducing the element to an impedance. If the curve spends only a little time crossing the initial region (or amplitude x hysteresis is enough to push it out around from center), the average will read high. Only when much time is spent near zero (both AC and DC bias approaching zero), do we read the initial permittivity as such.
