An AR(1)-GARCH(1,1) model without external regressors is 
\begin{aligned}
y_t &= \mu_t + u_t, \\
\mu_t &= \varphi_1 y_{t-1}, \\
u_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\
\varepsilon_t &\sim i.i.d(0,1). \\
\end{aligned}
It specifies the **entire conditional distribution** of the variable: its conditional mean, conditional variance, and density (its location given by the mean and its scale determined by the variance): $y_t\sim d(\mu_t,\sigma_t^2)$.

If you include regressors in the **conditional mean** model only (the equation for $\mu_t$), for different values of regressors you get different fitted and predicted values of $\mu_t$. Hence, you get different point forecasts (location shifts) but the same variance (and density, adjusted for the location) forecasts around the different points.

If you include regressors in the **conditional variance** model only (the equation for $\sigma_t^2$), for different values of regressors you get different fitted and predicted values of $\sigma_t^2$. Hence, you get different variance (and thus density) forecasts (scale shifts) around the same point forecasts.

If you include regressors in both the **conditional mean** *and* the **conditional variance** models, for different values of regressors you get different fitted and predicted values of $\mu_t$ *and* $\sigma_t^2$. Hence, you get different point forecasts (location shifts) *and* different variance forecasts (scale shifts) around the different point forecasts. So the density both shifts *and* scales due to the effects of regressors.