At human scales Matter and energy behave as if space was 3-dimensional. That does not mean space is 3-dimensional.
This may seem pedantic, but a lot of assumptions brake down when you look at tiny or high energy things. So, it's really important to mark down all your assumptions.
The mathematics that model space fit conveniently in 3 dimensions without becoming unmanageably complex.
However, by employing additional dimensions, particularly dimensions in which the basis vector multiplied by itself is a product other than itself, you can sometimes simplify the math that describes portions of the universe to a shocking extent. For instance, using geometric algebra with one dimension that squares to positive one and three that square to negative one, Maxwell's Equations reduce to "nabla field_bivector = free_space_permeability * speed_of_light * current_vector".
Sometimes, introducing special-purpose dimensions with interesting geometries and constraints upon the multidimensional representation makes certain types of math easier. For instance, conformal representation employs two dimensions that square to zero to represent the origin and infinity, and regular space is represented as a curved subset of the hyperdimensional space. Otherwise complicated operations become simpler, as a translation and rotation through dimensions that do not exist that almost coincidentally lands the result right back on the constrained hypersurface that represents 3-dimensional space.
When we invent these extra dimensions for the purposes of doing the math, we have no good way of knowing whether they are entirely imaginary or just undetectable and inaccessible to us at our current level of technology. If the universe had a round dimension with a diameter of less than the Planck distance, we would not be able to detect it, because we can't measure a distance that small. But maybe certain properties in the particle zoo can be more easily explained using an angular phase value on a tiny round dimension perpendicular to everything else.
Anthropic arguments such as this are good for examining reasons why spacetime with a different number of dimensions might not support interesting universes, but they fail to ask answer the question why we have a spacetime with 3 dimensions.
Some non-string theory attempts at formulating quantum gravity have started without an assumption of space-time and attempted to make it an emergent property, and given 3-dimensional space's quite unique properties that feels like it might ultimately be a better course of reasoning.
Couple that with Max Tegmark's ideas about multiverses. The idea is that all possible combinations of dimensions exist as a universe. We see 3+1 dimensions because this is the one combination that's most likely to develop life.
Consider the outside of a straw. The surface is a curved two-dimensional space which is bounded in one direction only by the length of the straw, but is quite small in the other (the circumference of the straw). If the straw is very long and very thin then it may appear to be essentially 1-dimensional, but the surface is still two dimensional.
Nice example. Wouldn't there be easily measurable evidence of this? Imagine a flattened caterpillar like creature living on such a straw. Unwittingly, one day it might wrap itself round the straw many times. Looking back, seeing itself turned round the straw in this way, wouldn't it see slices of itself rather than a whole?
Well, how can you go on forever without falling off the edge of the Earth? The surface of the Earth two-dimensional, and approximately flat... but there's small positive curvature, so you're continuously wrapping around to a different part of the space. The surface of the Earth can be meaningfully understood as a two-dimensional surface that possesses finite dimension.
A three-dimensional space can be close to flat but have positive curvature as well (or negative curvature, for that matter). Some proposals give our universe positive curvature, rendering its space finite, though still stupidly-big. (I'm not aware of there being a final word on the subject, though). And if different dimensions can have different curvatures, some of them could be much smaller than others.
My understanding is that we have conducted experiments to measure the curvature of the universe, and come up with answers that are withing the experimental error of being flat. Without any theoretical reason why the universe should be flat, it is still possible that the universe is curved, but to slightly for us to detect, however, but the simplest interpretation is to say that the universe is flat.
Of course, if we a assume a multiverse with universes having regularly distributed, positive curvatures, then, the size of the universe would grow asymptotically as the curvature approached flat, so, statistically, life is more more likely to arise on the flatter universes.
Is it possible, or in any way related, that a small but non-zero curvature could be responsible for the accelerated expansion of the universe? I don't know, it's hard to picture. But I'm thinking the idea that you can look at a sphere as a 2 dimensional space that curves and eventually wraps around such that things going opposite directions on its surface eventually can run into each other again.
That would probably be more easily measured though. I don't know.
A small non-zero curvature would correspond to a non-zero cosmological constant, which is precisely what seems to be driving the accelerated expansion of the universe (there are other explanations but this is the simplest one).
Infinite is a big word :) . As explained above, the extra spacial dimensions would need to be compact, which implies them being finite in size. Perhaps the simplest example of compact manifold is the circle: you can imagine one extra dimension to be closed on itself. Going around the circular dimension one circumference length, you will find yourself where you started. The size of the dimension then refers to the characteristic size of the circle, i.e. the circumference.
