Longitudinal waves can propagate through a medium as they move parallel to the direction of the wavefront...like sound, as they displace the medium they travel through in the direction of travel. Transverse waves move perpendicular to the direction of propagation, so cannot transmit through a medium like a gas or liquid (there's no way to generate motion in that direction) ...they can move across its surface, like a pond ripple. They can transmit through solids, though (think of earthquakes or vibrations in a pot when you bang it).
Here's a couple of site that'll help you out....
http://hyperphysics.phy-astr.gsu.edu...nd/tralon.html
http://www.matter.org.uk/Schools/Con...ransverse.html
http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
Looking at how waves propagate through a medium, I think I can see what the textbook is getting at. A longitudinal wave can pass through a 1 dimensional surface as it can readily propagate the wavefront by displacing the medium in the direction of travel. It can also do this through 3 dimensions. However, in 2 dimensions a wave can move up and down and/or side to side, with the motion of the wave being perpendicular to the direction of propagation. Hence no net displacement in the direction of propagation. So, flatlanders live in a very silent world
They could infer something was happening if they measured the displacement through a 1 dimensional surface, but all they'd be getting is a series of points moving past their measuring equipment and no information being carried. However, if one of us were to lift "Prof. Flatland Scientist" out of his/her space and into ours, he/she would hear him/herself scream for the first time!!
That could be disturbing as the flatlander would have no experience of hearing sounds and the mere thought of hearing a noise would probably be (initially, at least) rather frightening. To him/her, they'd appear to be coming from everywhere, including from inside of him/her!!!
Then again, being from a silent world, they may not have the ability to produce or hear sounds, so the experience would be a moot point.