von Neumann's approach was to define a "universal constructor", with a programmable "arm" for altering cells. He came up with a complicated set of cellular automaton rules which allow such a universal constructor. Then he came up with a "program" for the constructor which, when executed, would create a copy of the universal constructor and of the program.

This was all meant to be in analogy to the physical world, where such a constructor could be used to build almost anything, perhaps operating at the atomic-scale.

It turns out that we can get much simpler self-replicators if we (a) ignore the "universal constructor" ability, and only focus on self-replication; and (b) choose cellular automaton rules which make self-replication easier, rather than using cells to encode "programs". This approach gives things like Langton's Loops https://en.wikipedia.org/wiki/Langton%27s_loops

The self-replicating property described in the article seems like an extreme form of this approach: the rules are so tailored for replication that they can't do anything else.

Of course, making such simplifications sort of defeats von Neumann's original purpose, since such highly-tailored rules are less "realistic", and hence don't particularly help us build physical self-replicators; and even if we could, they wouldn't be particularly useful without the "universal constructor" ability.