This shows the $2$-chain
$$ d_3(abcd) = bcd - acd + abd - abc =
bcd + adc + abd + acb.
It consists of all the faces of the tetrahedron, oriented as
indicated by the circular arrows.
If you focus on any one edge, you will see that the two adjacent
circles rotate in opposite directions. Thus, in $d_2(d_3(abcd))$
the edge will appear twice, with opposite directions, which will
cancel out. Because of this, we have $d_2(d_3(abcd))=0$.