I am reading about the quantile function, but it is not clear to me. Could you provide a more intuitive explanation than the one provided below?

Since the cdf $F$ is a monotonically increasing function, it has an inverse; let us denote this by $F^{−1}$. If $F$ is the cdf of $X$, then $F^{−1}(\alpha)$ is the value of $x_\alpha$ such that $P(X \le x_\alpha) = \alpha$; this is called the $\alpha$ quantile of $F$. The value $F^{−1}(0.5)$ is the median of the distribution, with half of the probability mass on the left, and half on the right. The values $F^{−1}(0.25)$ and $F^{−1}(0.75)$ are the lower and upper quartiles.