I guess I have found another different proof of the fact I have asked. Could you check whether it is correct? Thanks
Lemma If $(M_t)_{t\geq 0}$ is a continuous local martingale, then $|M_t| < \infty$ a.s. for every $t\geq 0$.
Proof of the lemma: Suppose for sake of contradiction that $\exists s$ such that $P(|M_s|=\infty)>0$. If $(T_n)_n$ is a reducing sequence for $M$, then $\exists n$ such that $P(T_n > s)>0$. Thus $P(|M_{T_n \wedge t} |=\infty)>0$, but $M_{T_n \wedge t}$ is in $L^1$, since it is a martingale. $\Box$
Now, if we take $X_t$ continuous local martingale we have that also $X^2_t$ is a continuous local martingale and $|X^2_t| < \infty$ a.s. for every $t \geq 0$. (see comments)
The lemma applied to $X_t$ gives that $|X_t| < \infty$ a.s. for every $t \geq 0$ and thus $|X_t|^2 < \infty$ a.s. for every $t \geq 0$.
Finally, also $X^2_t-[X]_t$ is a continuous local martingale and hence we have $|X^2_t-[X]_t| < \infty$. From these facts follow that $[X]_t < \infty$ a.s. for every $t \geq 0$.