Von Neumann's crucial insight is that part of the replicator has a double use; being both an active component of the construction mechanism, and being the target of a passive copying process. This part is played by the tape of instructions in Von Neumann's combination of universal constructor plus instruction tape.
The combination of a universal constructor and a tape of instructions would i) allow self-replication, and also ii) guarantee that the open-ended complexity growth observed in biological organisms was possible. The image below illustrates this possibility.
This insight is all the more remarkable because it preceded the discovery of the structure of the DNA molecule by Watson and Crick, though it followed the Avery-MacLeod-McCarty experiment which identified DNA as the molecular carrier of genetic information in living organisms. The DNA molecule is processed by separate mechanisms that carry out its instructions and copy the DNA for insertion for the newly constructed cell. The ability to achieve open-ended evolution lies in the fact that, just as in nature, errors (mutations) in the copying of the genetic tape can lead to viable variants of the automaton, which can then evolve via natural selection.