Wightman quantum field - Interpretation

The smearing (test-) function is not a relic of the mathematical description, but a key ingredient of the theory. To quote from the fathers Wightman and Streater (PCT, Statistics, and all That):

It was recognized early in the analysis of field measurements for the electromagnetic field in quantum electrodynamics that, in their dependence on a space-time point, the components of fields are in general more singular than ordinary functions. This suggests that only smeared fields be required to yield well-defined operators. For example, in the case of the electric field, $\mathcal{E}(x,t)$ is not a well-defined operator, while $\int dx ~ dt ~ f(x) \mathcal{E}(x,t) = \mathcal{E}(f) $ is.

Another quote comes from BLT (Introduction to Aximatic Field Theory, 1975):

We define a quantum (or quantized) field as an operator- valued tensor distribution. Such a definition corresponds better to the real physical situation than the more familiar notion of a field as a quantity defined at each point of space- time. Indeed, in experiments the field strength is always measured not at a mathematical point $x$ but in some region of space and in a finite interval of time. Such a measurement is naturally described by the expectation value of the field as a distribution applied to a test function with support in the given space-time region.

It is also worth noting that classical fields are also distributions.

I like to think of it this way: We can only measure something with finite spatial resolution and for a finite time. So any experiment only measures an average over a small spacetime region. This is basically
$$ \phi(f)=\int \phi(x)f(x) d^4x $$ for some compactly supported smooth function $f$. This basically is what the distribution definition is doing. For technical reasons (we like Fourier transforms) people prefer Schwartz class to compactly supported test functions, but I doubt that it makes much difference to the physics.