Let's consider Joseph Fourier's concept of where an object can be broken down into a number of components and reconstructed from them. To be correct his concept was an infinite series of components but in practice a practical limit has to be imposed. If we take a note from a musical instrument and feed the sound into a frequency/spectrum analyser it will show the fundamental frequency of the note and number of harmonics. Each frequency being at different amplitudes. Attempts to create electronic music used these ideas. Remember the Moog Synthesizer and electronic organs of the 1970's.
If you did your own research into jpeg compression at this stage of the processing you would be told the 8 x 8 block is fed into a forward Discrete Cosine Transformation (DCT) process so that the image is transformed from a spatial domain representation to a frequency domain representation.
Most of you would just said what the Hell !!! This is perhaps the most confusing of all steps in the process and hardest to explain.
For a spatial domain think of an object in space with 3 dimensional coordinates
For frequency/time domain think of a time line with the slowest objects at one end and the fastest the other end. In fact what you would see on the display of a frequency/spectrum analyser.
So in effect we have taken a 8 x 8 picture element and fed it into a frequency/spectrum analyser to measure the strength of certain frequency components.
The output of the forward Discrete Cosine Transformation (DCT) process is a set of 64 values. Each value is the strength of each frequency component which start from a steady level (DC - average of all pixels) and each step is an increase in frequency in the x and y direction in the 8 x 8 block.
Still not easy to visualise.
So I managed to find a diagram showing the 64 frequencies. In my mind I find it easier think of these frequencies as a set of patterns and the DCT process is finding the strength of each pattern in the 8 x 8 block.
Therefore when it comes to decode the jpeg and reconstruct the 8 x 8 block it is easier to think of it as we need strength 4 for pattern 1, strength 8 for pattern 2, strength 0 for pattern 3, strength 12 for pattern 4 and so on. Almost paint by numbers
Just out of interest the DCT process is related to the fast Fourier transform.