For a real signal, the magnitudes of the Fourier transform bins tell what are the amplitudes of the sinusoidal components that an infinite tiling of the signal frame can be decomposed into. The phases of the bins encode translation of the sinusoids along their direction, like so:
The translation can be decomposed into a series of component translations one along each of the axes. The axes of an image are spatial axes, and in video the third axis is temporal. The component translations have magnitudes inversely proportional to the frequencies indicated by the multidimensional index of the bin. For example, you can easily spatially translate an image in the frequency domain by adding to all phases (in a polar form representation of the bins) a function of form $af_x + bf_y$ where $a$ and $b$ are constants that give the amount of translation and $f_x$ and $f_y$ are the vertical and horizontal frequencies (bin indexes).
Translation of a sinusoid of zero frequency is a bit of an oxymoron. With real FFT its phase is fixed to zero.