First here, there is a typo in the Dvoretzky paper: there must be $-1+\frac{1}{2}t^2E[X_{n,k}^2|F_{n,k-1}]$ instead of $-1-\frac{1}{2}t^2E[X_{n,k}^2|F_{n,k-1}]$ there. Otherwise, the inequality will not be true in general. Indeed, let, for brevity, $X:=X_{n,k}$, $F:=F_{n,k-1}$, $E_F Z:=E(Z|F)$, and $c:=\epsilon$. Suppose, e.g., that $X$ is independent of $F$, $P(X=1)=P(X=-1)=1/2$, and $c=1$. Then the erroneous inequality becomes
\begin{equation}
|\cos t-1-t^2/2|\le\tfrac16\,|t|^3,
\end{equation}
which is false for small enough $|t|$, since $|\cos t-1-t^2/2|\sim t^2$ as $t\to0$.

So, we actually need to show that
$$|E_F\exp(itX)-1+\tfrac12\,t^2E_F X^2|\le \tfrac16\,|t|^3 E_F|X|^3\,1_{|X|\le c}+t^2E_FX^2 1_{|X|>c} $$
given that $E_F X=0$.

By Taylor's theorem with the integral form of the remainder,
\begin{equation*}
|e^{ix}-1-ix-(ix)^2/2|\le|x|^3/6 \tag{1}
\end{equation*}
and
\begin{equation*}
|e^{ix}-1-ix|\le x^2/2
\end{equation*}
for real $x$.
The latter inequality also implies
\begin{equation*}
|e^{ix}-1-ix-(ix)^2/2|\le|e^{ix}-1-ix|+|(ix)^2/2|\le x^2/2+x^2/2=x^2. \tag{2}
\end{equation*}

Let $r(x):=e^{ix}-1-ix-(ix)^2/2$ and write, in view of (1) and (2):
\begin{multline*}
|E_F\exp(itX)-1+\tfrac12\,t^2E_F X^2|
=|E_F r(tX)|\le|E_F r(tX)\,1_{|X|\le c}|+|E_F r(tX)\,1_{|X|>c}| \\
\le\tfrac16\,|t|^3 E_F|X|^3\,1_{|X|\le c}+t^2E_FX^2 1_{|X|>c},
\end{multline*}
as desired.