Well one has (among others) three ways of seeing the 'flag variety'. The collection of all length $n$ flags: $$\mathcal{F}=((0)\subsetneq V_1\subsetneq V_2\subsetneq \cdots V_n=K^n),$$
where these $V_i$ are $i$-dimensional $K$-vector spaces.

We can see these as affine spaces, and hence realise this inside of the projective variety:
$$Gr(0,n)\times Gr(1,n)\times\cdots Gr(n,n).$$

Alternatively, one can consider the algebraic group $GL(n,K)$ acting on the standard flag $$(\{0\}\subset \langle e_1\rangle \subset \langle e_1,e_2\rangle \subset \cdots \langle e_1,\dots,e_n\rangle)$$

So that the flag $GL_n/B_n\cong \text{Fl}_n$. We have yet another realisation though... Let $\mathfrak{B}$ be the collection of all Borel subgroups of $G$. One ca prove that all Borel subgroups are conjugate, and hence for any $B'\in\mathfrak{B}$, we can get to whatever standard Borel we fix by $x^{-1}B'x=B$ i.e. $xB$ is a fixed point of the $B'$ action on $G/B$ (by left multiplication). If one proves the normalizer theorem where any Borel is self-normalizing, $G=N_G(B)$, one can show that for any 'other' fixed point $yB$ of $G/B$ we have $$xBx^{-1}=B'=yBy^{-1},\implies y^{-1}x\in N_G(B)=B,…

in general i think the idea that you can count things by doing intersection theory on a suitable moduli space pretty neat anyway. you may already know about schubert calculus - where you can use the cohomology ring/chow ring of the grassmannian to deduce enumerative results, e.g. given four general lines in $\mathbb{P}^3$ there are two lines passing through all 4 of them,

or e.g. that there are 27 lines lying on a general cubic surface in $\mathbb{P}^3$ (computing some chern class of some vector bundle over the grassmannian) - of course this is true for all smooth cubics, and there is a di…