Lemma 2.1.2 (Hoeffding's lemma). Let $X$ be a scalar variable taking values in an interval $[a, b]$. Then for any $t>0$, $$ \mathbf{E} e^{t X} \leq e^{t \mathbf{E} X}\left(1+O\left(t^2 \operatorname{Var}(X) \exp (O(t(b-a)))\right) .\right. $$ In particular $$ \mathbf{E} e^{t X} \leq e^{t \mathbf{E} X} \exp \left(O\left(t^2(b-a)^2\right)\right) . $$
$f=O(\phi)$ when $x\to a$ means $|f|<K\phi$ for all $x$ differing from but sufficiently near to $a$.
What would the corresponding $a$ be in the statement of Hoeffding's Inequality? Would $a=0$? We could then interpret this as, as you send $t$ sufficiently close to zero, there is some $K$ such that $\exp(O(t^2(b-a)^2))<\exp(K(t^2(b-a)^2))$?
From that estimate, I think we are more interested in the limit $t\rightarrow\infty$ than $t\rightarrow 0$. Hence, $\exp(O(t^2(b-a)^2))$ grows as something like $\exp(C(t^2(b-a)^2))$ for some constant $C>0$ as $t\rightarrow\infty$. But for these statement to be precise, they should always state which limit ($t\rightarrow 0 \text{ or } \infty$) they consider.