What this means is that as light travels across the universe, objects get further apart. Nowhere does it say that 'the fabric of spacetime itself' stretches, but the effect is similar to stretching a rubber sheet. Remember that. The effect is 'similar' to stretching a rubber sheet, yet it doesn't require that there actually be something there. That is perhaps the most confusing part. If nothing is there, what's bending, stretching, etc? Nothing. It is a product of geometry.
But what is geometry? Put simply, it is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space
http://en.wikipedia.org/wiki/Geometry
Normal geometry you learn in school is called Euclidean Geometry, and it is only concerned with 'flat' space. Certain rules apply here, such as the fact that all angles of a triangle must add up to equal 180 degrees. However, non-Euclidean Geometry is different. It turns out that if you have a non-Euclidean space, a triangle may not have each angle add up to 180 degrees! The common example is how the equator and two lines of longitude can form a triangle with the north pole. It's a triangle all right, it's got 3 sides and 3 angles, but all three angles add up to more than 180 degrees.
http://en.wikipedia.org/wiki/File:Triangles_(spherical_geometry).jpg
http://en.wikipedia.org/wiki/Non-Euclidean_geometry
However, while the example above puts the triangle on the surface of the Earth, nothing says that a surface is required. Space itself can simply work that way, regardless of whether or not it is made up of 'something'.
I think the main reason this is such a hard concept to grasp is that the effects of curved spacetime are not visible on our scale here on Earth. If everyone could see the effects of curved spacetime in their everyday lives there would be no question. No one asks why three angles of a triangle always add up to 180 degrees in flat space. No one thinks it's weird. It's simply accepted. Then when they hear that this geometry is wrong, they can't accept it. And that's what it boils down to. The geometry we all know and love is plain wrong. It's just accurate enough to use on our scale. (So I've been told by chronos or marcus or someone here at least)