Someone had once said you could be terrified in space, but you could not be worried there. It was perfectly true. (8.31)
Heywood Floyd thinks that space is a place for big emotions. We haven't been to space, so we can't say if that's accurate—but it's certainly true that 2001 prefers big, huge, emotions to smaller more contemplative ones.
Part 2, Chapter 12
It was barely audible, yet it stopped them dead, so that they stood paralyzed on the trail with their jaws hanging slackly. (2.12)
It's not exactly clear if the man-apes are filled with awe, if they're semi-hypnotized, or just confused. The slack-jawed stare comes across as something of a parody of watching television—especially since the man-apes eventually watch images in the slab. Awe is kind of watching a screen—it's like the man-apes in the novel are watching the man-apes in the film of 2001 (and are amazed).
So perhaps these visitors had come from the stars—yet that was even more incredible…Speculation was a waste of time: he must wait until there was more evidence. (12.26)
Floyd thinks about the vastness of space and time…and then chastises himself and says he needs evidence. This is a nice thumbnail illustration of how the novel works. It gives you giant, speculative, preposterousness to swallow (evolving man-apes! Energy aliens!) and then follows that up with serious, slow-moving descriptions of technical details. The details and the plea for evidence are supposed to make the awesome ridiculousness more real—even though it might make us yawn sometimes.
Part 2, Chapter 14
Some immaterial pattern of energy throwing off a spark of radiation like the wake of a racing speedboat, had leaped from the face of the Moon, and was heading out toward the stars. (14.8)
There is lots of whooshing from earth and out towards the distance in 2001. You start here and rush out to there; the distance and the speed are both meant to be amazing and impressive.
Part 3, Chapter 19
It was an eerie sound, for it had nothing to do with Man; it was as lonely and as meaningless as the murmur of waves on a beach, or the distant crash of thunder beyond the horizon. (19.8)
Bowman is here listening to background radiation from space. The sense that space is bigger than humans is meant to create a sense of insignificance and impressiveness. At the same time, the point of the novel is that there are vast forces out there that do, unexpectedly, care about humans. Humans are part of a vast, cosmic plan, tied into that distant crash of thunder in unexpected ways.
Part 4, Chapter 25
And then, almost as if in response to his plea, Poole waved back. For an instant, Bowman felt the skin prickling at the base of his scalp. The words he was about to call died on his suddenly parched lips. For he knew that his friend could not possibly be alive; and yet he waved. (25.31-32)
This is a zombie moment; the dead wave and come back to life. It's one of the few incidents where awe and amazement are linked up to horror (as they are in the work of writers like H.P. Lovecraft and H.G. Wells).
Part 4, Chapter 30
"We do not know if, out on the moons of Saturn, you will meet with good or with evil—or only with ruins a thousand times older than Troy." (30.23)
Heywood Floyd compares the possible alien ruins to the ruined city of ancient Greece. It's hard to imagine that anyone would actually say something like this; it sounds like something you'd read from a script. But it's thrown in there for the same reason the novel is called "A Space Odyssey"—to reference other old, important, impressive stories and make the events here seem important and impressive.
Part 5, Chapter 36
"It's just like the thing you found on the Moon! This is TMA-1's big brother!" (36.6)
Size is used as a way to provoke awe and amazement. The novel first has the little slab that teaches the man-apes; then the bigger slab on the moon, then the biggest slab, each more amazing than the last.
Part 5, Chapter 39
"The thing's hollow—it goes on forever—and—oh my God!—it's full of stars!" (39.20)
Bowman is wowed by the Star Gate. We would be too, understandably. This could be read as a description of the way Clarke sees sci-fi itself. It is vast, it is full of stars, and isn't that just shockingly nifty?
Part 6, Chapter 43
He did not even attempt to grasp the scale of the inferno toward which he was descending. (43.2)
Again, this seems like Clarke telling you how he would like you to read his book, or like you to see his book. All this vastness of space and time—do not even attempt to grasp it. Just leave your jaw open a little like the man-apes, and be stunned.