Let me start by some general considerations. In a theory with massless scalars it is possible that these scalars acquire non-trivial expectation values. The space of the possible expectation values is the moduli space $M$ of vacua of the theory. The kinetic term of the scalars in the low-energy effective action around a given vacua, or equivalently the two points functions of the scalar fluctuations around the vacua, defines a natural metric on the moduli space $M$. If the massless scalars are the only massless degrees of freedom then the low energy description of the theory is the sigma model of target $M$. But if there exists other massless degrees of freedom the low energy description is in general more complicated.

For a $\mathcal{N}=2$ $4d$ gauge theory, the low energy description at a generic point of the moduli space of vacua is an abelian gauge theory. In particular if the abelian gauge group is non trivial it is something more complicated than a sigma model with values in the moduli space of vacua.

After compactification on a circle we obtain a $\mathcal{N}=4$ $3d$ gauge theory. At low energy, at a generic point of the moduli space of vacua, we obtain again an abelian gauge theory. The key point is that in three dimensions an abelian gauge field is dual to a scalar field. Thus all the (bosonic) massless degrees of freedom can be seen as scalars and so the low energy effective description of the theory is the sigma model of target the moduli space $M$ of these scalars. This $3d$ sigma model has $\mathcal{N}=4$ supersymmetries ($8$ real supercharges). This implies that $M$ is naturally Hyperkähler. The easiest way to see that is maybe to reduce to two dimensions: it is classical that a $2d$ sigma model has $\mathcal{N}=(4,4)$ supersymmetries if and only if the target is Hyperkähler (for more informations and references see the answer to this question: http://physicsoverflow.org/23966/why-are-complex-structures-important-in-physics ). The idea is that the $\mathcal{N}=(4,4)$ $2d$ supersymmetric algebra has a $SO(4) \sim SU(2) \times SU(2)$ $R$-symmetry rotating the four supersymmetries which also rotates three complex structures $I,J,K$ on $M$.