[EDIT: I rewrote the first couple of paragraph, because I realized a better way to say what I had in mind.]
Depth is a sort of dimension. Perhaps not the most obvious, but one that works well in many situation.
In general we count dimension by chains and the main difference between Krull dimension and depth is about the same as the difference between Weil divisors and Cartier divisors.
For simplicity assume that we are talking about finite dimensional spaces. Infinite dimension can be dealt with by saying that it contains arbitrary dimensional finite dimensional spaces where we may substitute "Krull dimension" or "depth" in place of "dimension".
I always usually think of Krull dimension as going from small to large: We start with a (closed) point, embed it into a curve, then to a surface , until we get to the maximal dimension. (Let's assume we are talking about finite things here)However, for comparing to depth it is probably better to think of it as going from large to small: Take a(n irreducible) Weil divisor, then a(n irreducible) Weil divisor in that and so on until you get to a point.
In contrast, when we deal with depth we go from large to smalltake Cartier divisors: We start with the space itself (or an irreducible component), then take a a(n irreducible) hypersurface, then the intersection of two hypersurfaces=a hypersurfaces (such that it is a "true" hypersurface in each irreducible component this condition corresponds to the "non-zero divisor" provision)=a codimension $2$ complete intersection, and so on until we reach a zero dimensional set.
So, this picture is based on a naive-ideal geometric picture in which these two notions give the same number of steps. I would say that the geometric meaning of Cohen-Macaulay is that it is a space where our intuition about these two notions giving the same number is correct. I would also point out that this does not mean that necessarily all Weil divisors are Cartier, just that one cannot get a longer sequence of subsequent Weil divisors than Cartier divisors.