Your intuition is just what we all start with as we view these questions pictorially. Our mental picturing apparatus seems to have some hard-wired limitations, specifically to assume a Euclidean structure. Kant even claimed that we were not able to think of space in other terms, an unfortunate choice given the important role non-Euclidean geometries was to play in physics. In formal math courses, we learn some other ways of thinking about curvature, defined purely in terms of distances between points in a single space, without reference to embedding in some bigger Euclidean space. That's not to say that curved spaces can't be embedded in higher-dimensional flat space, just that the only real point of discussing that would be if it had some observational implications. Otherwise, you're just tacking some hypothetical dimensions onto the observed world.
Your 2D critters can measure various effects of the curvature of their space. If it's very mild, they can say they live in a flat space with some gravity. If it's stronger and they want a reasonably compact description, they'll say they live in a curved space. If all the curvature fits a theory in which the causes are already observable in the 2D space, they gain nothing by hypothesizing bowling balls or even higher dimensions. If there are features (e.g. analogous to our accelerating cosmic expansion) that have no observed source in the 2D space, they may want to explore ideas about higher-dimensions to find a theory that explains these phenomena too.
I know this is hard to wrap your mind around, and I have no illusions of having explained it well, so feel free to keep following up.
(published on 06/02/11)