This answer has four parts 1) , 2), 3), 4) . 1) is about complex structures and two dimensional sigma models. 2) is about complex structures and string theory from the worldsheet point of view, 3) is about complex structures and string theory from the spacetime point of view, 4) is about the relation between 1),2),3).
1) The notions of Kähler and Hyperkähler manifolds naturally appear in the context of two dimensional (2d) sigma model. A 2d sigma model is a 2d quantum field theory whose bosonic field is a map from the 2d spacetime to some target space $M$ and with a Lagrangian density of the form
$\mathcal{L}= \frac{1}{2} (g_{\mu \nu} + B_{\mu \nu}) \partial_i x^\mu \partial^i x^\nu$
where $g_{\mu \nu}$ is a Riemannian metric on $M$ and $B_{\mu \nu}$ is an antisymmetric tensor called the B-field. I will assume that the B-field is zero, which is the relevant case for the apparition of Kähler and Hyperkähler structures. The inclusion of a non-zero B-field would lead to a "twist" of these notions.
So, for every Riemannian manifold, $(M,g)$, there exists a bosonic sigma model of target $(M,g)$. One can shows that this sigma model has a natural supersymmetric extension with $N=(1,1)$ 2d supersymmetry: one has to include fermionic fields and a four fermions interaction coupled to the curvature tensor of $M$. So, for every Riemannian manifold $(M,g)$, there exists a supersymmetric $N=(1,1)$ sigma model of target $(M,g)$. In such a theory, the supersymmetric variation of the bosonic field $x^\mu$ under a supersymmetric variation of parameter $\epsilon$, a 2d Majorana spinor, is of the form $\delta x^\mu = \bar{\epsilon} \psi^\mu$ where the $\psi^\mu$'s are the fermionic fields.
A natural question is: under which condition does the supersymmetric $N=(1,1)$ sigma model of target $(M,g)$ admit extended supersymmetries ? The answer is that this theory admits an extended $N=(2,2)$ 2d supersymmetry if and only if $(M,g)$ admits a Kähler structure, and it admits an extended $N=(4,4)$ 2d supersymmetry if and only if $(M,g)$ admits a Hyperkähler structure. The Kähler case was first treated by Zumino in
http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?197909068
and then reviewed with the Hyperkähler case is in this paper of Alvarez-Gaumé and Freedman
http://projecteuclid.org/euclid.cmp/1103919984 ;
Basically, the complex structure appears as follows. A further supersymmetry transformation of parameter $\epsilon$ has necessarely the form $\delta x^\mu = \bar{\epsilon} J^\mu_\nu \psi^\nu$ for some tensor $J^\mu_\nu$. It is possible to show that the condition that the supersymmetry squares to a translation implies that $J^2 = -1$, i.e. that $J$ is an almost complex structure, and that the condition of invariance of the Lagrangian of the theory under this transformation implies that $J$ is covariantly constant with respect to the metric $g$. An covariantly constant almost complex structure is exactly the definition of a Kähler structure. Details can be found in the paper by Alvarez-Gaumé and Freedman.
2) A superstring theory is obtained by coupling some 2d supergravity theory to some 2d conformal field theory satisfying some appropriate condition. I will consider type II superstring (type I and Heterotic are similar: only consider "one half" of the worldsheet theory to be supersymmetric). A type II superstring background is obtained by coupling the $N=(1,1)$ 2d supergravity theory to some 2d $N=(1,1)$ superconformal field theory (SCFT) of central charge $c=15$. If one is interested in semi-realistic model, one generally take this CFT to be the product of a 2d $N=(1,1)$ free SCFT of central charge $c=6$, which corresponds to a sigma model to four flat non-compact dimensions, by a "internal" 2d $N=(1,1)$ SCFT of central charge $c=9$.
A natural way to construct a 2d $N=(1,1)$ theory is to consider the supersymmetric sigma model of part 1) with some target $(M,g)$. The $c=9$ condition implies that $M$ has to be of real dimension $6$. So we are looking for 6 dimensional Riemannian manifold $(M,g)$ such that the associated $N=(1,1)$ 2d supersymmetric model is a CFT. At one loop order, the CFT condition implies that $(M,g)$ has to be Ricci flat. All known Ricci flat Riemannian manifold are of restricted holonomy, in particular, all known $(M,g)$ of real dimension 6 have their holonomy group contains in $SU(3)$, i.e. are Calabi-Yau, and in particular Kähler (the existence of Ricci flat manifolds not with restricted holonomy is an open mathematical problem). By the discussion of part 1), the corresponding supersymmetric sigma model with such target has an extended $N=(2,2)$ 2d supersymmetry.
Here the conclusion is that $\mathbb{R}^{1,3} \times X$ with $X$ a 3-Calabi-Yau is a natural background for superstring theory, essentially because 3-Calabi-Yau's are the only Ricci flat manifolds of real dimension six that we know (maybe there are others but we don't know).
3) Superstring theory lives in 10 dimensions. On ten flat dimensions, it has some number of spacetime supersymmetries ($N=2$ 10d i.e. 32 real supercharges for type II and $N=1$ 10d i.e. 16 real supercharges for type I and Heterotic). To construct semi-realistic model, one considers superstring theory on a spacetime of the form $\mathbb{R}^{1,3} \times X$ where $X$ is a compact real dimension 6 Riemannian manifold. In order to be a consistent string background, $X$ has to satisfy some conditions, it is the subject of part 2). Here, ignoring these conditions, we adopt a spacetime point of view and try to impose a different constraint: the preservation of some of the original 10d spacetime supersymmetries in 4d after compactification on $X$. It is easy to see that preserved supersymmetries correspond to covariantly constant spinors on $X$. The existence of a covariantly constant spinor on $X$ implies that the holonomy group of $X$ a priori contained in $SO(6) \sim SU(4)$ is in fact contained in $SU(3)$, i.e. that $X$ is Calabi-Yau.
The condition for having 4d low energy supersymmetry is essentially phenomenological. It was (and probably still is) thought that the most likely low-energy completion of the Standard Model is given by a 4d $N=1$ supersymmetric theories. This amount of supersymmetry is precisely what is obtained by compactifying Heterotic string theory on a Calabi-Yau of holonomy strictly $SU(3)$.
4) Relation between 1)-2)-3). 1) and 2) are clearly related because perturbative string theory is 2d gravity coupled to some 2d sigma model. 1) is of course much more general: there exists many more general 2d sigma models than 2d sigma models satisfying conditions required by string theory (conformal, dimension 6) and for example many 2d sigma model with "higher" dimensional target are of the greatest interest in the study of various quantum field theories. The relation between 2) and 3) is that the 3-Calabi-Yau condition appears both as a condition for $N=(2,2)$ 2d worldsheet CFT and as a condition for 4d spacetime supersymmetry. In fact, this relation is not a coincidence, it can be explained, see for example the section 18.5 of the volume 2 of Polchinski's book on string theory, but in a non-completely trivial way: as always, the relation between worldsheet supersymmetry and spacetime supersymmetry in string theory is not completely obvious.
I have previously written that 1), i.e. the 2d sigma model point of view, was kind of the most general framework to explain the relevance of special geometric structures in quantum field theory/string theory. I should remark that for M-theory, there is no worldsheet, no 2d sigma model, and the only known explanations for special geometry in M-theory, such as $G_2$ holonomy compactifications, come from spacetime arguments as in 3).
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