The motivation comes from topology. Suppose you have a map of topological spaces f:X --> Y. You want to form the homotopical version of Y/X, so instead of identifying X to a point immediately, you do so gradually, so you form Y union_f CX, where X is the cone on X, and you glue the base of the cone to Y by the map f.

So from the point of view of the cells, the cells of the mapping cone on f in dimension n are

the cells of Y in dimension n

the cells of X in dimension n-1 (cross the 1-cell in the cone direction).

To do this with cochain complexes, you ought to form

Y in degree n

direct sum

X in degree n+1 (because you're using cochain, not chain)

The differential also comes from the topological motivation.

The motivation comes from topology. Suppose you have a map of topological spaces $f:X \to Y$. You want to form the homotopical version of $Y/X$, so instead of identifying X to a point immediately, you do so gradually, so you form $Y \cup_f CX$, where X is the cone on X, and you glue the base of the cone to Y by the map f.

So from the point of view of the cells, the cells of the mapping cone on f in dimension n are

the cells of Y in dimension n

the cells of X in dimension n-1 (cross the 1-cell in the cone direction).

To do this with cochain complexes, you ought to form

Y in degree n

direct sum

X in degree n+1 (because you're using cochain, not chain)

The differential also comes from the topological motivation.