Benjamin Antineau's answer needs a minor correction. For $X= spec (k)$, we have $CH^n (X, n) = K^M _n (k)$, not $CH^{2n} (X, n)$. Indeed not too many is known, but it is a very interesting subject (at least for me) to pursue. A good start would be Burt Totaro's paper 'Milnor K-theory is the simplest part of K-theory' or something similarly titled, where you can find the cubical version of it. Using $\mathbb{A}^1$-invariance with some spectral sequence arguments, one can prove that the above "simplicial version" and "cubical version" are isomorphic, thus equivalent.

Going back to Peter Arndt's question about 'intuition', the easiest one would be to look it as an algebro-geometric version of singular homology theory.

For instance, when $X$ is a topological space, a singular $n$-simplex is given by a continuous map $s: \Delta ^n \to X$. We collect their formal finite sums over the integers, and apply some simplicial formalisms. That's how we get the singular complex.

When $X$ is a variety, the problem is bad, even if we take $\Delta^n$ to be the algebraic n-simplex. One problem would be that there aren't enough morphisms of varieties $s: \Delta^n \to X$ to begin with. So, a way out is to take all "correspondences", i.e. closed subvarieties in the product space $\Delta^n \times X$. One problem that still persists here is that, to be able to apply the simplicial formalism, one has to have a good intersection property of correspondences with the faces of $\Delta^n$, but by taking all algebraic cycles, one may not get it. Consequently, we put conditions such as proper intersection with all faces.

That's why we define things in this way.

Regarding the question of what kernel/image does: it is difficult to explain everything, but the easiest case might worth paying attention: for instance, $z^i (X, 0)$ is the codimension i algebraic cycles on $X$, and the boundary map $z^i (X, 1) \to z^i (X, 0)$ by definition gives the rational equivalence of cycles on $X$. In this way, from the cokernel for instance, we recover the Chow group.