It depends on what you mean by "how".

The Chinese knew how to magnetise needles, knew that the compass didn't point to astronomical north, used them for navigation, by the late 11th century at the latest. The sources are discussed in Needham, Science and Civilisation in China, Vol 4, Part I.

The the compass points north (or south) because the Earth itself is a magnet was explained in William Gilbert's De Magnete, 1600.

All of that was in the absence of knowing how magnetic attraction actually worked. Explanations were typically Cartesian, in terms of emanations, atmospheres, streams of subtle particles, etc. A major step towards modern electromagnetic theory was by Aepinus, who Newtonianised the theories of electricity and magnetism. This could be taken as an explanation of the "how" (but not the "why"), in the same sense that Newton explained how gravitation worked.

Next, Maxwell's theory of electromagnetism, 1865. This is still "how", not "why". The relativistic theory of electromagnetism (Einstein, 1905; Minkowski, 1907) can be understood as a key step towards "why" by uncovering the relationship between electric and magnetic forces.

The next key step (and pretty much the current understanding) would be the development of quantum electrodynamics (QED), for which Feynman, Schwinger, and Tomonaga were awarded the Nobel Prize in Physics in 1965, which provided a quantum field theory explanation of magnetic attraction.

But it can still be reasonably claimed that we don't know why magnetic attraction exists, and that while mathematical models developed since Gilbert have given us better understanding of "how", not "why", Gilbert (or the earlier Chinese) already understood "how".

There's always room for further answers but I discussed the medieval/early modern history of conceptions of how compasses worked here - hope that helps!