Synthesized answer
The phrase "higher and higher" refers to aspirations towards greater dimensions [1, 2]. The narrator, speaking to a Sphere, suggests that just as the Sphere is superior to Flatland forms and combines many Circles into One, there must be a Being above the Sphere who combines many Spheres into a Supreme Existence [1]. This implies a progression of existence through more spacious and dimensionable spaces [1].
The passages indicate that "higher" can also refer to dimensions that are not easily perceptible. In Flatland, "height" is a Third Dimension that is "extremely small" and not easily perceptible to the inhabitants [3, 5]. The narrator's "upward course" and "corporal ascent" are linked to exploring these increasingly dimensionable regions, from the "blessed region of Four Dimensions" to the Fifth, Sixth, and Seventh Dimensions [2].
Synthesized from the book passages below. Chat with the book on Feynman for follow-up.
From the book
this ill-timed impertinent request? And what mean you by saying that I am no longer the Perfection of all Beauty? _I_. My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself. As you yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there…
s really Thoughtland, then take me to that blessed Region where I in Thought shall see the insides of all solid things. There, before my ravished eye, a Cube moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own—shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger at the…
eyes of his blind servant to the invisible presence of a Third Dimension, though I saw it not. Let me recall the past. Was I not taught below that when I saw a Line and inferred a Plane, I in reality saw a Third unrecognized Dimension, not the same as brightness, called “height”? And does it not now follow that, in this region, when I see a Plane and infer a Solid, I really see a Fourth unrecognized Dimension, not the same as colour, but existent, though infinitesimal and incapable of measurement? And besides this, there is the Argument from Analogy of Figures. Sphere. Analogy!…
for more knowledge. Doubtless we cannot _see_ that other higher Spaceland now, because we have no eye in our stomachs. But, just as there _was_ the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there _was_ close at hand, and touching my frame, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has…
sible. Surely you must recognize this? _I_. I must indeed confess that I do not in the least understand your Lordship. When we in Flatland see a Line, we see length and _brightness_. If the brightness disappears, the Line is extinguished, and, as you say, ceases to occupy Space. But am I to suppose that your Lordship gives the brightness the title of a Dimension, and that what we call “bright” you call “high”? _Stranger_. No, indeed. By “height” I mean a Dimension like your length: only, with you, “height” is not so easily perceptible, being extremely small. _I_. My Lord, your…