In the chapter Three is Company:
For a short way they followed the lane westwards. Then leaving it they turned left and took quietly to the fields again. They went in single file along hedgerows and the borders of coppices, and night fell dark about them. In their dark cloaks they were as invisible as if they all had magic rings. Since they were all hobbits, and were trying to be silent, they made no noise that even hobbits would hear. Even the wild things in the fields and woods hardly noticed their passing.
In the same chapter, the movie scene with the Ringwraith plays out, though slightly differently; the Ringwraith stops and seems to be sniffing them out, then suddenly trots away:
The other two ran quickly to the left and down into a little hollow not far from the road. There they lay flat. Frodo hesitated for a second: curiosity or some other feeling was struggling with his desire to hide. The sound of hoofs drew nearer. Just in time he threw himself down in a patch of long grass behind a tree that over-shadowed the road. Then he lifted his head and peered cautiously above one of the great roots. [...] From inside the hood came a noise as of someone sniffing to catch an elusive scent; the head turned from side to side of the road. [...] At that moment the rider sat up, and shook the reins. The horse stepped forward, walking slowly at first, and then breaking into a quick trot.
The Hobbits get caught again while loudly singing, at night when the Ringwraiths can see, and only the intervention of the Elves saves them:
The hoofs drew nearer. They had no time to find any hiding-place better than the general darkness under the trees; Sam and Pippin crouched behind a large tree-bole, while Frodo crept back a few yards towards the lane. It showed grey and pale, a line of fading light through the wood. Above it the stars were thick in the dim sky, but there was no moon.
Once more the desire to slip on the Ring came over Frodo; but this time it was stronger than before. So strong that, almost before he realized what he was doing, his hand was groping in his pocket. But at that moment there came a sound like mingled song and laughter.
In short, like the quote says, this isn't a supernatural ability - they're just very good at moving quietly and quickly hiding. And yes, it does seem to be in part to tie it in as historical - in the same paragraph you quote, Tolkien says they are still around today, and then explains why seldom see them.
Even in ancient days they were, as a rule, shy of ‘the Big Folk’, as they call us, and now they avoid us with dismay and are becoming hard to find. They are quick of hearing and sharp-eyed, and though they are inclined to be fat and do not hurry unnecessarily, they are nonetheless nimble and deft in their movements. (Prologue)