Cosmologists use the term “flat” a little differently than most people think of it, hence the common confusion. Basically, the curvature of the universe (which we’ll get to in a second) is determined by something called the critical density fraction, or mass-density fraction, or various other names, or just Omega. Omega is the ratio of the amount of matter in the universe to the amount of matter needed to close the universe (i.e. stop expansion). (it gets a bit more complicated with inflation, cosmological constants, and all that, but it’ll do for our purposes)
If Omega > 1, then the universe is positively curved; if Omega < 1, it is negatively curved, and if Omega = 1, it is flat.
This is usually explained via 2-D analogies, since most of us can’t visualize into a 4th dimension. Picture a sphere. If you draw a triangle on its surface, the angles will add up to more than 180 degrees. Now picture what’s usually referred to as a “saddle”; here a triangle’s angles add up to less than 180 degrees. Now picture a perfectly flat, or Euclidian surface.
Space has similar curvature, but in an extra dimension. Just as it’s possible for a 2-d person on the sphere to measure the angles of a triangle and determine that the space he lives in is curved, so should such effects be seen in the universe if space is curved. The problem with figuring this out observationally is that space is very, very big, and there’s no telling how large of scales we would have to look at to detect curvature if it existed–if someone painted a perfect 1-meter square on the sidewalk, could you figure out that the earth was a sphere because the angles totaled more than 360 degrees?
So, you see, the term “flat” refers more to the way things behave geometrically than an actual picture of the universe.
As mentioned in the article you quoted (by the way, you would not believe how excited my cosmology prof was on the morning of the day the BOOMERANG results were released, having seen them a little ahead of time), the geometry of the universe also has something to do with its fate. If Omega > 1 (positive curvature), then at some point expansion will stop, then reverse, and everything will come back together in a “Big Crunch”. If Omega < 1, the universe will continue expansion forever (and become a rather dreary place after a while, I might add). If space is truly flat, as BOOMERANG has pointed to pretty strongly, then expansion will stop after an infinite amount of time (subtle but important distinction from “expand forever”). (again, it’s a bit more complicated when there’s more than just matter/gravity at work, but this is the basic idea)
Most cosmologists have long been rooting for a flat universe; basically, it’s the cleanest and most elegant solution to all the various equations. Naturally, many have pointed out that this is a silly reason to believe something. But BOOMERANG is a big boost to the idea, and another (I believe space-borne) experiment in a few years should make it even clearer (or shoot it down).
Of course, providing this explanation for flatness/curvature to the question you asked leaves open the question of what space is expanding into, one which I don’t think I can provide a very good explanation for (especially since I’m about to get kicked out of here). This page has some quick answers towards the top.