Let us assume we are interested in some (forward) rate $F_t=F(t,T)$ which we assume is log-normally distributed:
$$\text{d}F_t=\sigma F_t\text{d}W_t$$
However, we observe market rates can in practice be negative. In order to circumvent this issue, we would like to use shifted log-normal dynamics:
$$\sigma(F_t+s)\text{d}W_t$$
where $s>0$ is the shift. We therefore define the shifted rate $F_t^s=f(F_t)=F_t+s$, which has the same dynamics than $F_t$ (apply Itô's Lemma to $f(F_t)$):
$$\text{d}F_t^s=\sigma F_t^s\text{d}W_t$$
Yet we know the solution to the above SDE:
$$\begin{align}
F_t^s&=F_0^s\exp\left\{-\frac{\sigma^2}{2}t+\sigma W_t\right\}
\\
&=(F_0+s)\exp\left\{-\frac{\sigma^2}{2}t+\sigma W_t\right\}
\end{align}$$
Thus if $s>|F_0|$ where $|\cdot|$ is the absolute value we ensure $F_t^s>0$ for all $t$. Note that we of course have that:
$$\begin{align}
F_t &= F_t^s-s
\\
&= -s+(F_0+s)\exp\left\{-\frac{\sigma^2}{2}t+\sigma W_t\right\}
\end{align}$$
So that we can recover the proper rate once the shifted rate has been simulated. In particular note that the forwards are preserved under the forward measure $\mathcal{T}$, that is:
$$E^\mathcal{T}\left(F_t\right)=E^\mathcal{T}\left(F_t^s-s\right)$$
It is the shifted rate $F_t^s$ that will not turn negative, not the rate $F_t$ itself: after all the reason shifted log-normal models were introduced is because we were observing negative rates on the market. A shifted log-normal model allows to represent rates that can be negative while preserving the preexisting modeling infrastructure based on log-normal dynamics.