The fringing pattern is caused by thin-film interference within the CCD. The signal received in a pixel will be proportional to the light falling on it, multiplied by a sensitivity, but then some extra signal is added or subtracted which depends on how much of the incoming light is at particular wavelengths that are affected by the interference (i.e. the whole broadband signal does not contribute, only the flux at certain wavelengths).
The fringing pattern is therefore an additive modulation of the sky background, which in general, for broadband filters (e.g. the i filter is 100 nm wide), has both a broadband component (scattered sunlight, moonlight) and a component at discrete wavelengths (airglow lines). Only the latter make a significant contribution to the fringing.
The fringing does not affect stellar photometry measurements, other than by introducing structure in the background that can be difficult to deal with when performing background subtraction. However, it does affect the images cosmetically and disrupts photometric measurements of extended objects.
It is not possible to remove perfectly because the amplitude (and also the shape) of the fringing pattern depend exactly on the spectral make-up of light illuminating the pixels and this will vary depending on (e.g. time of night, whether the Moon is up).
A first-order correction is to make a "fringe frame" by median stacking all the moonless night time exposures, after scaling them by the median sky signal (or the exposure time). A scaled version of this can then be subtracted from every processed image frame to remove the background component that is affected by fringing. It is hence referred to as an additive correction.
Twilight flat fields are used for an entirely different purpose. At twilight the sky background is close-to-uniform and has a broadband continuum spectrum that is dominated by scattered sunlight. It is therefore very difficult to see any fringing in it with broadband filters.
That is good, because the purpose of the twilight flats is to calibrate out the broad band sensitivity of each pixel by dividing by the flat field (hence referred to as a multiplicative correction, and you would normally multiply by a "balance frame" constructed by dividing a smoothed version of the median twilight flat by the unsmoothed version). The twilight flats can also be used to calibrate any large spatial scale sensitivity variations caused for example by vignetting. Twilight flats that do show evidence of strong fringing in broadband filters should not be used to remove pixel to pixel sensitivity variations. In those cases one usually resorts to "dome flats".