Ok, so let's work in the setting of a Noetherian local ring $(R, \mathfrak{m})$.  Then the Krull intersection says that $\bigcap_{n \geq 1} \mathfrak{m}^n = (0)$.  (You can remove the local hypothesis if you assume that $R$ is a domain, and then $\mathfrak{m}$ can be any ideal).

The easiest geometric interpretation of this statement that I can think of is something like the following.  

*There is no hypersurface passing through a point (or subvariety) of a variety $X$ with infinite order of vanishing through that point/subvariety.*

Or, 

*The only function which vanishes to arbitrarily high order at a point is the zero function.*

The interpretation should be pretty easy.  Given an $f \in R$, we can measure the order of vanishing of $f$ by asking what's the biggest power $n$ of $\mathfrak{m}$ such that $f \in \mathfrak{m}^n$ but such that $f \notin \mathfrak{m}^{n+1}$.  Given now a scheme $X$, go look at a stalk of some (possibly non-closed) point.

Obviously, this result shows up a lot when studying completion, so it has geometric applications as well.