I've Googled quite a few words from that sentence and still can't wrap my head round it. Can anyone ELI6?
Individual development (ontogeny) restates historical evolution (phylogeny). Haeckel’s biogenetic law says that as an embryo develops it will pass through its past historical states as a species. It's a long discredited 19th century scientific hypothesis.
Inverting the relationship would state that historical evolution is restating individual development OR in the other sense of inverting, they move in the opposite order to one another (Mathematicians: Euclid -> Topology, Kids: Topology -> Euclid).
Basically, kids build up knowledge of topology from everyday interaction with the world and then move onto Euclidean shapes with the knowledge of topology.
Mathematicians discovered it all in the opposite order, starting with Euclidean geometry and only later documenting topology.
Häckel came from a 19th century conception of “advancement” or “progres” — think of that absurd 19th century picture of various apes followed by a human, as if a human is somehow the “most evolved” of the animals. The idea of adaptation to the environment was alien (and apparently frightening) to most people in the West.
Separately: some people, like Spencer, who believe that “progress” has a direction, that “more advanced” societies have more universal knowledge that accumulates like stacking paper on a table, (and this was the challenge of Darwin) that humans are closest to god and all other creatures are inferior (you can see how this extends to a belief that there are different races (=breeds) of humans, both in genetics/god’s engineering and in culture).
Then tying the two together: you can look at the development of an individual human as an iterative, never ending and somewhat meandering process or you can say that you just have to transfer those stacks of paper, bottom to top, and then you’re done.
The connection between the two may sound absurdly tenuous but sad to say it is highly politicized and to this very day affects how schools are organized and funded, culture, politics, beliefs about right and wrong and how people should behave.