Calabi-Yau spaces have no effect on reality. However, they may shed some light on the traditional desire of humans to understand reality.
Calabi-Yau spaces are mathematical constructs which seem to have significant importance to superstring theory, which is one of the currently popular theoretical models to describe how the universe works. Superstring theory is aimed at reconciling the theories of quantum mechanics and general relativity.
The idea is that the universe consists of four-dimensional space-time and six additional dimensions which are looped into small structures called compact Calabi-Yau manifolds. This means that each point of space-time has associated with it a 6- or 7-dimensional knot of tangled geometry which is consistent with the parameters of a Calabi-Yau space. If the size of these tangles are small enough (say, less than about 10-16 centimeters), we would not currently have noticed them through physical experimentations.
A six-dimensional manifold having torsionless SU(3) structure is a Calabi-Yau manifold. Gravity acting in the compactified dimensions affects other non-gravitational forces such as electromagnetism. At this point, precision tests of gravity have only put upper limitations on the size of such hidden dimensions. Despite the attractiveness of superstring models, there is currently no experimental evidence that supports superstring models in favor of other theoretical models of reality.
Keep in mind that this is a MODEL of reality – it does not constrain reality in any manner whatsoever. In addition, at present superstring theory is not supported by any experimental evidence whatsoever.
Having set that straight, what is a Calabi-Yau space, and what does it have to do with how reality works? There are only a few dimensionalities for spacetime that allow the existence of massless photons. In simple string theory, the corresponding dimensionality is 26, while in superstring theory it is 10 or 11 dimensions to obtain a geometrically flat solution for the nontangled dimensions of spacetime.
So why are Calabi-Yau manifolds special? It turns out that no new forces appear from the small dimensions when they take the form of a Calabi-Yau manifold. The shape of the Calabi-Yau manifold has a huge effect on how the forces between strings appear in the 4-dimensional universe which we perceive.
In principle it is possible to deduce the nature of those extra dimensions by requiring consistency with the model, but this is not yet a practical possibility. It is also possible to extract information regarding the hidden dimensions by precision tests of gravity, but so far these have only put upper limitations on the size of such hidden dimensions.