'Hypostasis' is a philosophical term for the underlying reality of something, as opposed to the attributes or 'appearances' of the thing. For example, a red ball has attributes of being 'red' and 'round', but the thing-in-itself is something beyond those mere attributes.
'Hypostasise' (American: 'hypostasize') is a verb meaning to come to an understanding of the underlying reality of something. In other words, it's a valid philosophical question whether we can understand (hypostasize) the actual nature of a red ball, aside from the attributes that appear to our senses.
Basically, Ñanavira is saying that the conventional mind creates an image of the self by observing the attributes of the self in reflection: e.g., we do or say something, observe that which we've done or said, and then we think: "That is (or that isn't) the kind of person I am." Over time (and through many such reflections) we build up an image of our self, the same way we might build up an image of an actor by watching all of his or her movies.
In other words, the conventional mind thinks of itself in terms of the reflections of its own behavior that it sees occur, as though it were watching a movie of its own life and imagining itself as the protagonist. This is the 'play of reflections' that Ñanavira talks about. We do X (an act of speech or behavior) then evaluate ourselves in terms of the results. We roll a bowling ball down a lane and flatten all ten pins, then stop and think: "oh, yeah, I'm awesome, what a great bowler!". Then we rehearse what we did in our minds, trying to repeat the event and outcome. We think about what we did and didn't do, how we stepped and swung and released; we let our own self-reflections wash over us so that we can emulate our selves and create another strike.
That's the play of reflections: how we examine ourselves to correct what we did 'wrong' and solidify what we did 'right'.
Now, the natural inclination here is to presume that the self we have observed is the self itself. This is natural because that's what we assume about all external objects and entities. We see a horse behave in certain ways and thing: "that's how that horse is"; we see a rock display certain properties and think: "that's how that rock is". This is the moment of hypostasis. We see the way a thing behaves and we make assumptions (hypotheses) about the
internal essence of the thing: it's 'horse-ness' or it's 'rock-ness'. And likewise we see how we behave and assume that our accumulated behavior defines what we are. And thus the 'play of reflections' — our own observation of what we concretely do and say — reifies into an image of what we are. It's as though we see that every time we go to the movies we eat popcorn, and conclude that we are the type of thing that must always eat popcorn at movies. We have taken a behavior and come to a (not necessarily true) understanding of what we are.
There's nothing particularly odd about this. I'd really just an informal form of induction: we do X a number of times, and thus conclude that we have a disposition to do X. This is what bridges the first and second paragraphs. We watch ourselves do things on a daily basis, and over time we hypostasize (create an image) of our essential essence from what we observe. It's no different than deciding that some person is conniving while some other person is gullible, except that it's done through a process of self-reflection rather than other-observation. So all of these self-observations allow us to build up an image of ourselves — similar to the images we construct of others — and we quickly and easily conflate this self-image-object with the actual essence of ourselves. This is what leads to the concept of a soul: something that is simultaneously exterior to us (defined in our external behavior) and identical to us (essential to our being).
The arahant doesn't confuse the reflection with the essence. An arahant doesn't think: "I am the kind of person who always eats popcorn in theaters" just because s'he always has eaten popcorn in theaters. The induction doesn't apply, because the arahant is liberated from the karmic processes that dictate identity in that way.