The missing steps is that after assuming $p$ you have by $p\Rightarrow q$ that $q$ follows and then $q\lor s$ follows and by $(q\lor s)\Rightarrow t$ follows $t$. Now you have a contradiction because of $\
eg t$ so you can conclude $\
eg p$ instead.
In similar way you conclude $\
eg r$ and $\
eg p\land\
eg r$ follows.
A problem seem to be is that you're mixing up terminology. The deduction theorem states that you can prove a implication by assuming the premise and proving the consequence and by that is proving that the premise implies the consequence. Or formally:
$$\begin{align}\phi&\vdash \psi\\\ &\vdash \phi\Rightarrow \psi\end{align}$$
This is not that central to the problem, instead it's reductio ad absurdum which states that a statement can be proved by assuming it's opposite and proving a contradiction by which you can conclude the statement. Or formally:
$$\begin{align} \
eg\psi&\vdash\phi\\\ \
eg\psi&\vdash\
eg\phi\\\ &\vdash\psi \end{align}$$