What is meant by the term "value" of a scalar quantum field? During the slow roll of a scalar field, the scalar field is changing its value over time. But what is meant by the term "value" of a scalar field? Since the scalar field is quantized, I don't understand how the field itself can have a value. I read somewhere that it is some sort of average value of the field however that didn't really help me in understanding this.
 A: Your question is not specific to inflation, and really applies to any case where a bosonic quantum field behaves semiclassically due to macroscopically large occupation numbers. One very simple example of this is the Stark effect in quantum mechanics, where a Hyrodgen atom is placed in a uniform electric field. The atom is treated as a quantum mechanical bound state, but the electric field is treated classically. Of course, both systems are truly quantum, so why is it consistent to treat one quantum mechanically and the other classically? 
The reason is that the electric field has a macroscopically large expectation value (in whatever state is being considered), i.e. we may write
$$ A_{\mu} = \langle A_{\mu} \rangle + \delta A_{\mu} $$
where $A_{\mu}$ is the operator-valued gauge field, $\langle A_{\mu} \rangle$ is the expectation value, and $\delta A_{\mu}$ is a fluctuation term with zero mean. In the case where the electric field is very large and may be treated classically, this split makes perturbative sense because $\langle A_{\mu} \rangle^2 >> \langle \delta A_{\mu}^2 \rangle$, and so the fluctuations are small compared to the background value. If you wanted to correct the approach of treating the electric field classically, you could consider fluctuations in $\delta A_{\mu}$ in a perturbative expansion.
This same logic applies to more fancy systems such as Bose-Einstein condensates, which is essentially what the inflaton field is. In this case the bosonic field is a scalar, and once again it has macroscopic occupation numbers so that its fluctuations are suppressed and one may treat the system semi-classically. 
Getting back to your specific question about inflation, the value refers to $\langle \phi \rangle$, and this is what is changing over time. You could also ask how the various higher moments of the fluctuations vary over time, i.e. $\langle (\delta \phi)^n \rangle$, for $n \ge 2$.
