Without the bias voltage, there would be no electric bias field between either electrode relative to the diaphragm, against which bias fields the signal adds or subtracts in order to move the diaphragm bidirectionally.
The required bias voltage of an electric field headphone element (or speaker) is very much akin to the permanent magnet of a conventional electromagnetic loudspeaker -- both provide a fixed field against which the signal works to produce proportionate mechanical motion.
Whether this bias voltage requirement will eventually result in "phantom power"[1] on analog "headphone outputs" as it has [optionally] for professional analog microphone inputs[1] -- or small power cells and inverting electronics in the headphones or on the headphone wires (perhaps supporting other analog or DSP functions such as active noise cancellation, equalization for flatter response, or decryption of an encrypted digital headphone signal) -- may be up for the market's consideration in a few years.