A GARCHAn AR($r$1)-GARCH(1,$s$1) model without external regressors is \begin{aligned} y_t &= \mu_t + u_t, \\ \mu_t &= \dots \text{(e.g. a constant or an ARMA equation without the term $u_t$)}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dotsc + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.d(0,1). \\ \end{aligned}\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.