As others have mentioned, the reasons lie indeed in two-dimensional conformal field theory and in string theory.
The propagation of string on a compact Lie group $G$ is described by the Wess-Zumino-Witten model, whose dynamical variables are maps $g:\Sigma \to G$ from a riemann surface $\Sigma$ to $G$. The quantisation of that model is difficult in terms of $g$ (although see the 1988 papers of Gawedzki and Kupiainen, and also Felder, for a functional integral approach) and one instead chooses to quantise their currents, roughly the (anti)holomorphic components of the pullbacks $g^*\theta_L$ and $g^*\theta_R$ of the Maurer-Cartan forms on $G$. There is a natural action of two copies of the affine Kac-Moody algebra associated to $G$ on the WZW model which preserves the Poisson structure of the WZW model and gives rise to moment mappings which are, essentially, the currents. In other words, the Poisson bracket of the currents is that of two copies of the affine Kac-Moody algebra of $G$. Hence the quantisation naturally leads one to consider unitary, integrable representations of the affine Kac-Moody algebra. The first "modern" reference for this is a 1986 paper of Doron Gepner and Edward Witten String Theory on Group Manifolds; although there are pioneering papers of Halpern, Bardakci,... in from the late 1960s and early 1970s.
At a more abstract level, we can substitute the group $G$ by any (unitary) two-dimensional (super)conformal field theory with the right central charge. This idea of replacing the geometry by a conformal field theory used to be known as "strings without strings", since one loses the description of strings propagating in some geometry. In this context, it is important to have at one's disposal a number of unitary two-dimensional conformal field theories. The natural ones are those coming from from unitary representations of infinite-dimensional Heisenberg and Clifford algebras (so-called free fields) and unitary integrable representations of affine Kac-Moody algebras, but one can also consider constructions (e.g., orbifolds, coset constructions,...) which generate new unitary CFTs from these ones. The first "modern" reference for the coset construction is perhaps the 1986 paper of Peter Goddard, Adrian Kent and David Olive Unitary representations of the Virasoro and superVirasoro algebras.
Finally, I should say that although it's the affine Kac-Moody algebras which seem to have played the more important rôles thus far, there is also the emergence (in the context of M-theory) of more general Kac-Moody algebras. There's work on this in King's College London (West et al.), Brussels (Henneaux et al.) and Potsdam (Nicolai et al.). I'm not very familiar with this, though.