Given an action of the circle group $S = {\mathbb G}m$ on a smooth variety $X$, with isolated fixed points $X^S$, we can define a Bialynicki-Birula decomposition $$X = \coprod_{f\in X^S} X_f, \qquad X_f := \{ x \in X : \lim_{z\to 0} S(z)\cdot x = f }.$$ (What happened to the curly braces??) \}.$$Part of B-B's theorem is that each X_f is a copy of affine space. If Y \subseteq X is S-invariant, then Y acquires a similar decomposition, and Y_f = X_f \cap Y for each f\in Y^S \subseteq X^S (very easy to prove). Consider the embedding Y := GL_n/B = Flags(n) \to \prod_{k=1}^n Gr(k,n) \to \prod_{k=1}^n {\mathbb P}(Alt^k\ {\mathbb C}^n) =: X, where the second map is made of Plucker embeddings, and take S acting on {\mathbb C}^n by z\mapsto diag(z,z^2,z^3,\ldots,z^n), AKA the \check\rho coweight. Then its fixed points on each {\mathbb P}(Alt^k\ {\mathbb C}^n) are indexed by k-element subsets of 1\ldots n. So X^S is lists of subsets, and Y^S is increasing lists of subsets, or equivalently permutations. Ergo, there exists a decomposition of GL_n/B into affine spaces, indexed by permutations. (It's not obvious from this description that they are the B-orbits, but maybe that's okay, since more spaces have these S-actions than have finitely many B-orbits.) 1 While I basically agree with Kevin Buzzard that this is something to find in a textbook rather than on mathoverflow, I'll take the opportunity to give a totally nonstandard description, inspired by Shizuo Zhang's comment. Given an action of the circle group S = {\mathbb G}m on a smooth variety X, with isolated fixed points X^S, we can define a Bialynicki-Birula decomposition$$X = \coprod_{f\in X^S} X_f, \qquad X_f := { x \in X : \lim_{z\to 0} S(z)\cdot x = f }. (What happened to the curly braces??) Part of B-B's theorem is that each $X_f$ is a copy of affine space.
If $Y \subseteq X$ is $S$-invariant, then $Y$ acquires a similar decomposition, and $Y_f = X_f \cap Y$ for each $f\in Y^S \subseteq X^S$ (very easy to prove).
Consider the embedding $Y := GL_n/B = Flags(n) \to \prod_{k=1}^n Gr(k,n) \to \prod_{k=1}^n {\mathbb P}(Alt^k\ {\mathbb C}^n) =: X$, where the second map is made of Plucker embeddings, and take $S$ acting on ${\mathbb C}^n$ by $z\mapsto diag(z,z^2,z^3,\ldots,z^n)$, AKA the $\check\rho$ coweight. Then its fixed points on each ${\mathbb P}(Alt^k\ {\mathbb C}^n)$ are indexed by $k$-element subsets of $1\ldots n$. So $X^S$ is lists of subsets, and $Y^S$ is increasing lists of subsets, or equivalently permutations.
Ergo, there exists a decomposition of $GL_n/B$ into affine spaces, indexed by permutations. (It's not obvious from this description that they are the $B$-orbits, but maybe that's okay, since more spaces have these $S$-actions than have finitely many $B$-orbits.)