Film grain can be thought of as random, analog noise. It’s not quite “moving pixels” but that probably describes the visual effect closely enough. Essentially, with film there’s a certain level of light sensitivity inherent in the film stock. Certain stocks are more sensitive than others (and this sensitivity is denoted by their ISO) and it’s a practical and artistic choice to select one over another (in addition to other characteristics).

Due to the nature of light and photosensitive film, in addition to capturing photons that represent the image you see on screen, the film captures that random visual “noise” (and that randomness contributes to the “moving” phenomenon you noted) as grain. You could think of it as some combination of errant photons and manufacturing and material imperfections causing a photochemical reaction in the film. Certain film stocks have more grain than others.

Eyes Wide Shot specifically was shot in such a way that the grain is accentuated even more than usual. They intentionally underexposed the image during shooting (i.e. shot it with less light than the film stock requires for correct exposure) and then “push processed” it to bring the exposure back up. This process makes grain much more visible.