More figuratively, if you take any circular planar slice of the spatial part of an expanding spacetime such that all points in the plane have the same coordinate time and then evolve the slice forward in time or examine it over cosmological distances (meaning more than megaparsecs) the plane is always flat like a disc rather than curving like a bowl.
With the constraint that there are cosmological observers -- they just float with the metric expansion of space and always observe the largest-scale distribution of matter as isotropic and homogeneous: mostly meaning similar-looking galaxy clusters (and cosmic microwave radiation) along every line of sight -- then a Robertson-Walker metric has identical constant curvature everywhere, so it doesn't matter what planar slice you choose: you'll always see the same circular disc, spherical cap, or their equivalent on a negatively-curved plane. Moreover, it does not matter when we take the planar slice; the spatial curvature is constant in the whole spacetime. [1]
Thus, we have excellent evidence for the spatial flatness of the universe from the small variations in the blackbody temperature of the Cosmic Microwave Background, and the evidence improves as we compare those variations with galaxy clusters at different redshifts (and other measures of distance).
In his last paragraph (which provoked your question) roywiggins means that our universe is expanding and at cosmological scales matches the Robertson-Walker metric extremely well. If we look into the far future, the matter dilutes away enough that it will look very similar to de Sitter space (an empty expanding spacetime) rather than anti-de Sitter (an empty collapsing spacetime).
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[1] Just to clarify: we are talking about spatial curvature here, where we have sliced up the 4d spacetime into 3d spaces indexed by time. In an expanding universe, whatever the spatial curvature (even if it's exactly zero), spacetime curvature is large: freely-falling objects diverge with the expansion. The spatial curvature mostly deals with relative distortions of the shapes of similar galaxies (e.g. barred spirals, or ellipticals) at different distances. With zero spatial curvature, faraway barred spirals have the same shape as nearer barred spirals.
With positive spatial curvature, further away barred spirals would be less arm-y and more core-y than nearer ones, while with negative spatial curvature they would be more arm-y and less core-y. The light from the outer parts of the arms and the core are all emitted essentially as parallel rays. With positive spatial curvature, the rays converge, so the arm-rays get squashed towards the core-rays as they travel cosmological distances. With negative spatial curvature, the initially parallel rays diverge.
We don't see that kind of distortion in images like this, with a large number of faraway galaxies looking very similar to foreground galaxies like Andromeda (M31) https://en.wikipedia.org/wiki/Hubble_Ultra-Deep_Field