As argued by von Neumann, the measuring process has many properties that resemble those found in the theory of operator algebras. For instance, if you have an instrument, you can measure something, say the length of a table, to get a certain value $x$ within experimental errors. What you can now do is relabel the ticks of your instrument according to a certain function $f$. If you measure the same quantity you will now find $f(x)$ within some other error. Without loss of generality you can assume that the outcome of a measurement lies in a compact subset of the real line. The fact that one inevitably has to deal with experimental errors restrict the set of admissible functions to those that are uniformly continuous on this compact set, hence continuous. What you then have is something that resembles the spectral mapping theorem, that is
$$\sigma(f(O)) = f(\sigma(O))$$
where $O$ is an observable (note that I'm not assuming $O$ to be an operator yet, this will follow as a consequence of these heuristic considerations), and $f$ is the act of relabeling the ticks, whereas $\sigma(O)$ denotes for the time being all the possible outcomes of a measurement of $O$ (the so-called physical spectrum).
The way states are defined in this approach turns out to lead to a set, the set of admissible states, which has some mathematical properties. In particular this set, when equipped with a suitable topology, becomes compact and convex. The same holds for the state space (this name stems precisely from this fact) of a C*-algebra whereby one can identify the operation of measuring $O$ over a state $\omega$ as the evaluation of the linear functional $\omega$ over some "operator" $O$.
The above has then led to the axiomatisation of quantum (and classical as well, when one consider commutative settings only) mechanics in the following terms
Definition and axiom A physical system is a C*-algebra $A$ whose self-adjoint part is the set of observables and (a subset of) its state space is the set of all the physically admissible states for the system.
The "operators" that one usually deals with in quantum mechanics arise now as a consequence of the above axiom. A C*-algebra is an abstract object which becomes concrete when one considers a representation of it. Among all representations, the important ones are the irreducible representations. It turns out that a quantum mechanical systems with $n$ degrees of freedom that satisfy to the Heisenberg relations (i.e. the canonical commutation relations) generates a C*-algebra which is isomorphic to the C*-algebra of compact operators on an infinite dimensional separable Hilbert space. The theory of representation for such an algebra is such that there exists only one class of unitary equivalence of irreducible representation. Since one of these is the Schroedinger representation, this is essentially the unique one. The associated Hilbert space is $L^2(\mathbb R^n)$ with Lebesgue measure. This is how Dirac's formalism for quantum mechanics arises from just first principles, in a nutshell.