I am trying to understand the following definition:
Given probability space $(\Omega, \mathcal{F}, P)$ and an increasing sequence of sub-$\sigma$-fields $\mathcal{F}_0=\{\emptyset, \Omega\} \subseteq \mathcal{F}_1 \subseteq \cdots \subseteq \mathcal{F}_n = \mathcal{F},$ a sequence of random variables $X_0, X_1, \ldots, X_n$ (with finite expectations) is a martingale if for each $k=0,\ldots, n-1$ we have $E(X_{k+1} | \mathcal{F}_k)=X_k.$
I'm familiar with the concept of $\sigma$-algebras/fields, but I have never seen an expression like $E(X_{k+1} | \mathcal{F}_k)$. What does it mean to condition on $\mathcal{F}_k$? Is it viewing the function $X_{k+1}$ as restricted to $\mathcal{F}_k$? I have no intuition for what is trying to be conveyed in this definition...