While (I think) this answer will not provide the intuition behind, it hopefully will bring some insight.
One way to see where the white noise comes from in the $MA(q)$ representation is given by the Wold decomposition. The moving average $MA(q)$ and the autoregressive $AR(p)$ processes are specific cases of a general representation of stationary processes obtained by Wold.
Wold proved that any weakly stationary stochastic process, $z_{t}$, with finite mean, $\mu$, that does not contain deterministic components, can be written as a linear function of uncorrelated random variables, $a_{t}$, as:
\begin{array}
\ z_{t} & = & \mu + a_{t} + \psi_{1} a_{t-1} + \psi_{2} a_{t-2} + \ldots \\
& = & \mu + \sum^{\infty}_{i = 0} \psi_{i} a_{t-i} & ; & \psi_{0} = 1
\end{array}
Where:
$E(z_{t}) = \mu$
$E(a_{t}) = 0$
$Var(a_{t}) = \sigma^{2}$
$E(a_{t} a_{t-k}) = 0$ for $k>1$
We can write, $\tilde{z}_{t} = z_{t} - \mu$ and using the lag operator, then we have:
$$\tilde{z}_{t} = \psi (B) a_{t} \tag{1}$$
With $\psi(B) = 1 + \psi_{1} B + \psi_{2} B^{2} + \ldots$
Equation $(1)$ is the general linear representation of a non-deterministic stationary process. This representation is important because it guarantees that any stationary process admits a linear representation. In general, the variables $a_{t}$ make up a white noise process, that is, they are uncorrelated with zero mean and constant variance.
Taken from Andrés M. Alonso Fernández slides. More here
