What do loop groups and von Neumann algebras have to do with elliptic cohomology?

David Roberts mentioned in his comments the relationship

K-theory : spin group
TMF : string groups

Let me recommend the first 6 pages of my unfinished article for a uniform construction of $SO(n)$, $Spin(n)$, and $String(n)$, which suggests the existence of a similar uniform construction of $H\mathbb R$, $KO$ (or $KU$), and $TMF$. You'll see that von Neumann algebras appear in the construction. More precisely, von Neumann algebras appear in the definition of conformal nets. The latter are functors from 1-manifolds to von Neumann algberas.

For a summary of the conjectural relationship between conformal nets and $TMF$, have a look at page 8 of this other paper of mine (joint with Chris Douglas).

Loop groups yield non-trivial examples of conformal nets. Those conformal nets are related (conjecturally) to equivariant $TMF$.

There are various programs (which start with Segal's survey, I believe), all of which I know nothing about, to interpret elliptic cohomology classes in terms of von Neumann algebras, loop group representations, conformal field theories, ...

I can understand why a geometric interpretation of elliptic cohomology would be desirable, but it's mystifying to me why researchers in this area are concentrating on these specific objects.

You probably know most of this by now, but here are some thoughts. As usual I will be working at a very heuristic level throughout this answer.

Von Neumann Algebras. One place to start is the observation is that for $H$ an infinite-dimensional Hilbert space, the von Neumann algebra $B(H)$ has automorphism group the projective unitary group $PU(H)$. $PU(H)$ fits into a short exact sequence

$$1 \to U(1) \to U(H) \to PU(H) \to 1$$

and $U(H)$ is contractible by Kuiper's theorem; thus $PU(H)$ is a $B^2 \mathbb{Z}$, and $BPU(H)$ is a $B^3 \mathbb{Z}$. Hence $H^3(X, \mathbb{Z})$ is, in a suitable sense, a "Brauer group" of $X$ describing bundles of von Neumann algebras (isomorphic to $B(H)$) over $X$.

The significance of this observation is that $H^3(X, \mathbb{Z})$ is a natural cohomology group parameterizing twists of K-theory over $X$; equivalently, there is a natural map from $B^3 \mathbb{Z}$ to $BGL_1(KU)$. Given a bundle of von Neumann algebras over $X$, the corresponding twisted K-theory groups are something like the K-theory of module bundles of Hilbert modules over the bundle of von Neumann algebras, but don't trust me to have the specifics right here.

Now it's also known (see e.g. Ando-Blumberg-Gepner) that $H^4(X, \mathbb{Z})$ is a natural cohomology group parameterizing twists of tmf over $X$; equivalently, there is a natural map from $B^4 \mathbb{Z}$ to $BGL_1(tmf)$. If you could build a (higher) category which deloops von Neumann algebras in a suitable sense, you might hope to realize $BPU(H) \cong B^3 \mathbb{Z}$ as the automorphisms of an object in this category, and then families of those objects over $X$ could be a geometric avatar of these twists of tmf in the same way as above. I believe that conformal nets is explicitly intended to be such a delooping.

Loop group representations. This is the analogue of $G$-equivariant K-theory having something to do with the representation theory of $G$. One picture of tmf whose accuracy I can't comment on is that it should look heuristically like $K(ku)$, the cohomology theory presented by (Baas-Dundas-Rognes) 2-vector bundles. So $G$-equivariant tmf should look heuristically like $G$-equivariant 2-vector bundles, which over a point should look heuristically like representations of $G$ on (suitably dualizable) 2-vector spaces.

Whatever that means, such a thing ought to have a "character" which, rather than being a class function on $G$, or equivalently a function on the adjoint quotient $G/G$, is instead an equivariant vector bundle on $G/G$. Now $G/G$ looks heuristically like the classifying stack $BLG$ of the loop group $LG$, and hence an equivariant vector bundle on $G/G$ looks heuristically like a representation of $LG$. Freed-Hopkins-Teleman makes this precise. The non-equivariant version of this story is Witten's story about tmf having something to do with ($S^1$-equivariant?) K-theory of the free loop space $LX$.

Conformal field theories. This is the analogue of K-theory being presentable by vector bundles with connection. One way to think about a vector bundle with connection on a manifold $X$ is that it is a "$1$-dimensional topological field theory over $X$": that is, it assigns a vector space to a finite set of points equipped with signs in $X$ (the tensor product of either the fibers of the vector bundle or their dual depending on the orientation), and it assigns a map of vector spaces to every oriented cobordism between such points in $X$ (the tensor product of either the holonomies or the evaluation or coevaluation maps).

The historical motivation for generalizing this to $2$-dimensional conformal rather than topological field theories is, I think, to explain the modularity properties of the Witten genus. But again, don't trust me to have the specifics right here. (I guess it's $2$-dimensional topological rather than conformal field theories over $X$ that look more like $2$-vector bundles.)