In what sense might there be an "illusion of memory"?

(This is an extra scheduled post while I travel. It was written the weekend before Fluidity Forum.)

There's a line in E-Mailing the Lamas from Afar about an "illusion of memory" that I don't understand. An apprentice asks Ngak'chang Rinpoche about what rebirth would even mean given an assumption or doctrine that there is no such thing as a soul or an ongoing self. They say:

Apprentice: If the link between lives isn’t memory—and I guess it isn’t—then what is it?

Ngak’chang Rinpoche: The link is the empty thread—and that is there all the time. You could ask: ‘In what way will my next moment be ‘me’ if I have no memory of my present moment?’ What follows through from life to life is what follows through from moment to moment. We have an illusion of memory which allows us to feel as if some sort of continuity is taking place—but that is a fantasy. We imagine we are the same person simply getting older. That is also a fantasy. Every moment is just like starting over.

I'm not sure I follow. One idea might be that we are free to behave in any moment as a completely new person—even though for many reasons we mostly choose not to. The moment is "empty" or free, or open. The illusion of memory would then be an illusion that memory "means" that I am "the same person" and therefore have some propensity to continue behaving as I have before.

I wonder if there is more to this, though.