We use a Robertson-Walker (RW) metric to describe the universe at the largest scales because it does the job well. There are several components to the (vacuum) RW metric, and two are relevant to your question.
Let's take a planar ("equatorial") slice of the expanding universe at a given time. In that slice let's put two test objects that aren't interacting with each other in any way: the gravitational attraction is effectively zero, and there are no electromagnetic or other interactions between them. They also don't decay or radiate. In the flat Minkowski spacetime of Special Relativity these test objects would follow completely parallel worldlines eternally (to the infinite past or the infinite future): the spatial distances are the same in every slice.
One parameter of the RW metric controls the spatial distance between these test objects in the immediately preceeding and immediately following slices. That is the expansion function. In an expanding or contracting RW universe, these objects are spatially closer together in one immediately neighbouring slice and farther apart in the other. In the expanding case, the spatial distances are greater in each slice into the future, and smaller in each slice into the past. "Unslicing", if these objects could (without disturbing their trajectories) measure their distances using RADAR signals, the RADAR distances would always increase into the arbitrary future. Flat spacetimes do not expand: expansion is a manifestation of spacetime curvature.
Another parameter controls whether each slice is spatially flat. If a slice is spatially curved in an expanding RW spacetime, then optical distortions change the observed size of distant objects with the expansion. In practice, this would be encoded as a distance-dependency in the brightness-angle-redshift relationship observed in distant galaxies. This isn't required by current observations made in ultra-deep-field studies, so the universe cannot depart from spatial flatness by more than a tiny amount.
The third relevant parameter is the extent of each slice. In principle every slice can be spatially infinite, no matter where in the past or future the slice is, and that is what accords with observation. However, slices could be merely finite but very large, and there might be a function relating each slice's extent to its past-predecessor or future-successor. A "closed" universe is one in which [a] the spatial curvature discussed above is positive, [b] the slices are finite but very large, and [c] there is no boundary because the slice "wraps" around spherically or toroidally or in some other fashion, and [d] the expansion function decays into a contraction function. Any non-closed universe is "open" to some extent.
This is "punned" with the non-vacuum modelling of the Friedmann-Lemaître-Robertson-Walker expanding universe with various types of matter as a fluid "dust" embedded within the vacuum RW spacetime, wherein the matter in a non-closed RW universe in the sense of the previous paragraph is dense enough that it will eventually collapse. Our two test objects above would still have always-increasing RADAR distances while all the mutually-attracting charged matter that started around them collapses into ever denser structures.
Indeed, in the FLRW model the "dust" motes are galaxy clusters, which individually collapse in a Schwarzschild-like metric (typically one uses a Lemaître-Tolman-Bondi metric, since Schwarzschild is eternal, and LTB is a collapsing dust). However, at the galaxy-cluster scales they're like our idealized test objects: they don't interact much -- after clustered galaxies form they don't really push distant ones around with their emitted radiation, and clusters are far enough apart that the mutual gravitational attraction is basically zero. Coarsely, their RADAR distances always increase into the future. (More finely, clustered galaxies orbit around inside their clusters, so some galaxies (and bits of spinning galaxies) are moving away a bit faster and some slower than expansion carries them. This is the "peculiar motion" of galaxies, and star clusters within galaxies.)
So: spacetime curvature is large, because galaxy clusters were much closer together in the past. Spatial curvature is zero or close to it, because spiral galaxies have roughly the same basic shapes to them (not squashed or stretched) at all redshifts. The universe is open in the sense that in general widely separated galaxy clusters are not at any risk of recollapsing into each other: it is only peculiar motions of galaxy clusters that cause cluster-cluster collisions, like the Bullet Cluster. (Oh, would that such collisions were commonplace: it would provide lots of useful data! But most galaxy clusters are "Eulerian": they have an unexciting view of practically all other galaxy clusters receding from them exactly according to the expansion parameter of the Robertson-Walker metric.)
Can you elaborate? I was under the impression that the curvature is essentially zero and that the universe is open.