Rex asked:

"*As far as I am aware, theoretical physics is about finding mathematical models to explain observed physical phenomena. My questions therefore are:

What is the basic physical phenomenon/problem/question that vertex operators model?*"

Please have a look at the nLab at the pages that have already been mentioned:

ncatlab.org/nlab/show/vertex+operator+algebra and

ncatlab.org/nlab/show/conformal+field+theory

There is also a lot of material about QFT there.

Here is a quick and dirty and oversimplified introduction to some aspects of QFT:

In the Wightman framework we can think of quantum fields as operator valued distributions, that map test functions living on a given spacetime to (maybe unbounded) operators. Essentially selfadjoint operators represent observables that can be measured, in principle, by some experiment or device. Physicsists are mostly interested in systems that have interactions. We can think of elementary particles as localized excitations of quantumf fields that are solutions to specified wave equations (in the distributional sense). In order to have any interactions, these wave equations should have non-linear terms (actually you can define the notion of "free field" (a field with no interactions whatsoever) this way: A free field in the sense of physicists is a quantum field that is a solution of a linear equation).

Non-linear terms entail products of quantum fields, which are undefined, in general, because products of distributions are undefined, in general. The history of QFT is about the struggle of physicists to dodge this problem in one way or another. One way to dodge this problem is to introduce, as an axiom, so called "(associative) operator product expansions". This is a way to formulate, as an axiom, (handwaving:) that the "severity of the singularity of products of distributions" is under control.

**operator product expansion**

An operator product expansion (**OPE**) for a family of fields means that there is for all fields and all $z_1 \neq z_2$ a relation of the form
$$
\psi_i(z_1) \psi_j(z_2) \sim \sum_{k \in B_0} C_{ijk} (z_1 - z_2)^{h_k - h_i - h_j} \psi_k(z_2)
$$

Here $\sim$ means modulo regular functions. (See ncatlab.org/nlab/show/conformal+field+theory for an explanation of the notation.)

In a handwaving way, this axiom says that we assume we know something about the kind of singularity of the product of two fields, as their support comes closer and closer, and it is not so bad.

A rigorous interpretation of an OPE would interpret the given relation as a relation of e.g. matrix elements or vacuum expectation values.

An OPE is called **associative** if the expansion of a product of more than two fields does not depend on the order of the expansion of the products of two factors.
**Warning**: Since the OPE has no interpretation as defining products of operators, or more generally the product in a ring, the notion of associativity does not refer to the associativity of a product in a ring, as the term may suggest.

The axioms of vertex operator algebras are an **axiomatization of OPEs**. In this sense there is reason to expect that vertex operator algebras will play a key role, on one way or another, in a rigorous construction of interacting QFTs in four dimensions.