If you are familiar with PCA or SVD, you are already close to understanding a basic form of compression. SVD breaks down an m x n matrix into an nxn rotation matrix, an nxn diagonal scaling matrix, and a n mxn loadings matrix. The ordering of the new matrices is usually with the highest amount of variability described first. So if you take the first, say, 5 of the n dimensions, and only use them, you can reconstruct an approximation of the original matrix that uses approximately 5/n of the original storage.

PCA is often used in machine learning too, and there is such a deep connection between compression that is hard to make explicit or formalize.

The implementation linked here uses Vector Quantization instead:

https://en.wikipedia.org/wiki/Vector_quantization