This answer is a little bit redundant with the other two answers given so far, but here goes anyway.
It is easier to describe the real Grassmannian case. We can look at the Grassmannian $\text{Gr}(n,k)$, and suppose that $2k \le n$; if not then you can pass to the opposite Grassmannian. If $V$ and $W$ are two $k$-planes in $\mathbb{R}^n$, then there a set of $k$ 2-dimensional planes that are each perpendicular to each other and each intersect $V$ and $W$ in a line. Call the angles between these lines $\theta_1,\ldots,\theta_k$. Then in the connecting geodesic $V_t$ with $V_0 = W$ and $V_1 = V$, the angles are instead $t\theta_1,\ldots,t\theta_k$. This is an explicit description that is basically equivalent to Mariano's remark about invariant complements.
The complex version has the same system of angles, but complexified lines, 2-planes, and $k$-planes. The "angle" between two lines in a 2-plane can be defined as the geometric angle between their slopes plotted on the Riemann sphere. Actually these angles are twice as large as the angles in the real case in the previous paragraph, but that makes no difference.
I was vague on the positions of the planes. The orthogonal projection of $V$ onto $W$ has a singular value decomposition, and the singular values are $\cos \theta_1,\ldots,\cos \theta_n$. The orthogonal projection the other way is the transpose, or Hermitian transpose in the complex case, so it has the same singular values. The corresponding singular vectors are the lines $V \cap P_k$ and $W \cap P_k$. So the actual explicit work of finding the geodesic comes from solving a singular value problem, or equivalently an eigenvalue problem.