What London and Bauer are saying here is that quantum mechanics must be understood as not just a theory like any other that is, as about the world in some sense but as a theory of knowledge in itself, insofar as it implies a well-defined theory of the relation between the object and the observer. This represents a crucial difference from classical physics as it is usually understood. From the perspective of quantum mechanics, the relationship between the observer and the object being observed must now be seen as quite different from that which underpins the previous stance of naive realism, which is typically adopted with regard to classical mechanics and which holds that objects exist entirely independently of all observation and possess measurable properties, whether these are actually measured or not. That view must now be abandoned.
The core of London and Bauers text then represents an attempt to articulate the nature of that relationship between the observer and the object or system being measured.
London and Bauer radically depart from von Neumanns argument at a crucial juncture. In setting out the chain of correlations, from detector + system to observers body + detector + system, they do not stop at the consciousness of the observer but also include this in the overall quantum superposition.
It is this move that expresses in physical terms the phenomenological idea of the mutually dependent context of being, so that not just the body of the observer but their consciousness is also correlated, quantum mechanically, with the system under investigation.
How do we go from that correlation, manifested through the quantum superposition, to having a definite belief corresponding to our observation of a certain measurement outcome? Here, London and Bauer insist that
In other words, the transition from a superposition to a definite state is not triggered in some mysterious fashion by the consciousness of the observer and, as a result, Putnam and Shimonys concern regarding how consciousness can cause a definite state to be produced is simply sidestepped. Instead, what we have is a separation of consciousness from the superposition, leading to a new objectivity, that is, a definite belief on the part of the observer and a definite state attributed to the system.
This separation is effected, as London and Bauer explain, via

And, in a typed note inserted by London in his own copy of the manuscript, he wrote:
It is this characteristic and familiar act of reflection that cuts the chain of statistical correlations expressed by quantum theory as a set of nested superpositions, and keeps the twin phenomenological poles of those correlations namely consciousness and the world mutually separate. And so, on the one hand, the system is objectified, or made objective, in the sense of having a definite state attributed to it, and, on the other, the observer acquires a definite belief state through this objectifying act of reflection.