Let $\Gamma$ be a normal form game. An equilibrium $s$ of $\Gamma$ is a perfect equilibrium of $\Gamma$ if $s$ is a limit point of a sequence $\{s ( \eta ) \} _{\eta \downarrow 0}$ with $s(\eta)\in E(\Gamma,\eta)$ for all $\eta $ i.e. $s$ is perfect if there exist sequences $\{s(t)\}_{t \in\mathbb{N}}$ and $\{\eta(t)\}_{t\in \mathbb{N}}$ with $s(t)\in E(\Gamma,\eta(t))$ for all $t\in\mathbb{N }$, and such that $s(t)$ converges to $s$ and $\eta(t)$ converges to zero, as $t$ tends to infinity.
My question is: how do I get and precisely understand the switch from $$s(\eta) \text{ to } s(t)$$ ?
If I understand your question correctly, this is just notational density. You use $\eta$ as the perturbation (minimal tremble) of the stratgies. That is $E(\Gamma, \eta)$ is the set of equilibria of the game $\Gamma$ such that all strategies for all agents puts weight at least $\eta$ on each action (pure strategy). We call an equilibrium proper if it can be approximated arbitrarily well by such restricted games.
Your sequence $\eta(t)$ is just $\eta_1, \eta_2, ...$, is a sequential relaxation of the perturbation of the game (in the sense of action weights). What is needed then is a corresponding sequence of equilibria $s\big(\eta(t)\big) \in E\big(\Gamma, \eta(t)\big)$ that similarly converge to $s$. Think of it as approximation: if as a sequence of perturbations subsides to zero, we would wish that there is a sequence of equilibria of the perturbed games that converges (in the space of strategies) to the equilibrium of interest, $s$.
In action, consider the following game.
$$ \begin{array}{|c|c|c|} \hline & L & R \\ \hline T & 1,1 & 0,0 \\ \hline B & 0,0 & 0,0 \\ \hline \end{array} $$
Clearly $\{B,R\}$ is an equilibrium. It is not, however, perfect. This is because $R$ is a best response for the column player if and only if the row player's strategy plays $B$ with probability one. In any tremble, row puts at most probability $1-\eta$ on $B$, hence every equilibrium of every perturbed game with $\eta > 0$ puts weight $\eta$ on $R$ for column, hence as $\eta\to 0$, any sequence of equilibria of perturbed games must converge toward $\{T,L\}$, the unique perfect equilibrium of this game.