The dielectric properties of thin films are of paramount important in a variety of technological applications, and of fundamental importance for solid state research. In spite of this, there is currently no theoretical understanding of the dependence of the dielectric permittivity on the thickness of thin films. We develop a confinement model within the Lorentz-field framework for the microscopic Langevin-equation description of dielectric response in terms of the atomic-scale vibrational modes of the solid. Based on this, we derive analytical expressions for the dielectric permittivity as a function of thin film thickness, in excellent agreement with experimental data of Barium-Strontium-Titanate (BST) thin films of different stoichiometry. The theory shows that the decrease of dielectric permittivity with decreasing thickness is directly caused by the restriction in k-space of the available eigenmodes for field-induced alignment of ions and charged groups.