Why is the Complete Flag Variety an algebraic variety?
because of that identification we can put a variety (projective) structure on it...comes from projective structure of $\textrm{GL}(n,\mathbb{C})/B_n$.
there is another identification that this collection of complete flags can be thought of as inside product of $\mathbb{G}(1,n) \times \mathbb{G}(2,n) \times \cdots \times \mathbb{G}(n-1,n)$ where $\mathbb{G}(r,n)$ Grassmanian variety .