Why does capacitor value changes when capacitor is DC biased?
Murata has a pretty interesting explanation on its FAQ section about what is going on. I will quote the most relevant parts here for future reference.

The characteristic of change in capacitance according to the applied voltage is called "DC (direct current) bias characteristic."



The mechanism of DC bias characteristic

In the high dielectric constant capacitor type of monolithic ceramic capacitors, at present mainly BaTiO3 (barium titanate) is used as a principal component of high dielectric.
As shown below, BaTiO3 has a perovskite shaped crystal structure and above the Curie temperature it becomes a cubic shape with Ba2+ ions to the vertices, O2- ion to face center and Ti4+ ion in a body centered position.



At the Curie temperature (approx 125°C) or more, it has a cubic crystal structure, and below the Curie temperature and within an ambient temperature range, one axis (axis C) stretches and other the axes shrink and turn to a tetragonal crystal structure.

In this case, polarization occurs as a result of the unit shift of axially elongated Ti4+ ion crystal. This polarization occurs without applying an external electric field or pressure, and is known as "spontaneous polarization." As explained above, a characteristic that has a spontaneous polarization and a property of changing orientation of spontaneous polarization by an external electric field to reverse is called "ferroelectricity."

The reversal of the spontaneous polarization per unit volume is equivalent to relative permittivity. Relative permittivity is observed as a capacitance.
Without a DC voltage, spontaneous polarization can happen freely. However, when a DC voltage is externally applied, spontaneous polarization is tied to the direction of the electric field in the dielectric, and independent reversal of spontaneous polarization is inhibited. As a result, the capacitance becomes lower than before applying the bias.