What is the Johnson-Lindenstrauss lemma intuition in simple terms?

It states that high-dimensional points can be projected into low dimensions while approximately preserving distances.

The Johnson-Lindenstrauss lemma guarantees that a small set of points in high-dimensional space can be embedded into a much lower-dimensional space using random projections, while preserving pairwise distances within a small error. This works because random projections tend to distribute variance evenly across dimensions, avoiding bias toward any single direction.