Thanks to this question: Explanation of Lagrange Interpolating Polynomial, I have an intuition for what $\frac{x-x_i}{x_k-x_i}$ is doing in polynomial interpolation. That is, it is a kind of "on and off switch", enabling us to pass through specific points in our data or function.
However, I don't understand the expression, $\frac{x-x_i}{x_k-x_i}$, in any geometric sense. "The distance between some point $x$ and $x_i$, divided by the distance between $x_k$ and $x_i$" doesn't mean much to me.
That term is not very meaningful by itself, but the Lagrange polynomial associated to a node $x_k$, given by
$$L_k(x) = \dfrac{\prod_{i \ne k} (x-x_i)}{\prod_{i \ne k} (x_k-x_i)},$$ is a polynomial of degree $\leq n$ (if you have $n+1$ interpolation points) whose value is one on "its node" ($L_k(x_k)=1$) and zero over all others.
This function performs the mapping
$$x_i\to0,\\x_k\to1,$$ in a way creating "boolean" variables. When you multiply several such expressions, you get the effect of a logical and.