I believe that this is taken from homology theory in algebraic topology. Simplicical homology studies the maps from simplices (i.e. points, lines, triangles, tetrahedrons and so on in increasing dimensions) into some fixed topological space. An $n$-chain (or just chain if the dimension is implicitly understood) is a formal sum of such maps of $n$-dimensional simplices. For example, a $1$-chain is a set of curves in the space (or more precicely, a set of maps from the unit interval into the space).
A chain is called closed if the total boundary map (with orientation) is zero. I assume the reason is that in the one-dimension case you can take a closed form and stitch together the curves it represents into one or more closed curves. In the two-dimensional case you can stitch together the "triangles" to form spheres ("closed" discs) and so on.
In simplicical cohomology theory you study homomorphisms from chains into some group $G$. It can be shown that the simplicical cohomology theory developed from this is more or less equivalent to the de-Rahm cohomology defined with differential forms on manifolds. It would therefor make sense to transfer much of the terminology, including the word closed.