$\require{AMscd}$ Let $C$ be an pointed $\infty$-category (by which I mean quasicategory) admitting cofibers. For an object $X \in C$, why is $X \simeq cofib(X[-1] \to 0)$?
Here $X[-1]=\Omega X$ is an object such that
$\require{AMScd}$ \begin{CD} X[-1] @>a>> 0\\ @V b V V @VV c V\\ 0 @>>d> X \end{CD}
is a pullback square. If $C$ is stable, then that means this is also a pushout square, and so then I see why. But in Higher Algebra Lemma 1.1.2.10, C is just a pointed infinity category, not necessarily stable. I'm sure the answer is incredibly simple.
This is certainly not true for an arbitrary pointed $\infty$-category with cofibers. For instance, in the category of pointed spaces, this would mean that the canonical map $\Sigma\Omega X\to X$ is a weak equivalence for all (nice) pointed spaces $X$, which is very false (for a simple example, take $X=S^0$). However, the statement of Lemma 1.1.2.10 has an extra hypothesis: that the suspension functor is an equivalence on $C$.