What is the role of Freqency in Digital Images
You can take images as 2D discrete signals. The "time" in 1D signals is actually two spatial dimensions in images (2D signals).

You can measure "frequency" as well - imagine a line of white pixels with regular spacing. The spacing represents period $p$, and frequency is given by $1/p$. Hence the maximum frequency the discrete signal can contain is limited by the sampling rate.

When creating a digital image, sampling can cause new frequencies to appear due to aliasing. See Nyquist frequency.

Fourier and wavelet transforms have discrete variants to work with discrete signals - very simply said: integrals are replaced by sums.

Wavelets are more efficient in image compression, because they can deal with localized redundancies. Simply look of DFT of a grey square on black background and then on DWT of that square.

DWT will exploit transitions of the square shape in high fequency layer, and other layers will be almost clear. The sharp 1px transition would be best described with Haar wavelet without any remainder.

On the other hand, DFT or DCT will result in many cosine functions which summed together form the edge without visible ringing artifacts (Gibbs effect).

Wavelet have its artifacts too, but these are better localized, hence does not corrupt rest of the image (or rest of the macroblock in case of JPEG/DCT compression).