sigma algebra representing information

Question

Let $(\Omega, \mathcal{A},P)$ be a probability space, $\mathbb{F}=(\mathcal{F}_{n})_{n \in \mathbb{N}}$ a filtration on $\mathcal{A}$ and $(X_{n})_{n \in \mathbb{N}}$ a stochastic process adapted to $\mathbb{F}$. I don't understand how one can regard $\mathcal{F}_{n}$ as the available information at time $n$ for $n \in \mathbb{N}$. Furthermore, let $X$ be an integrable random variable. Can one explain $E[X \mid\mathcal{F}_{n}]$ descriptive as some approximation of $X$ having some information?