While studying differential geometry, I read this part of a proof and I didn't understand it. Given a $2$-manifold $\Omega$ and an interval $I=(-\epsilon, \epsilon)$, consider the cartesian product $M=\Omega\times I$. The exterior derivative on the algebra of differential forms on $M$ splits in this way: $d=d_{\Omega} + d_{I}$, where $d_{\Omega}$ and $d_{I}$ are respectively the exterior derivatives on $\Omega$ and $I$, and, called $t$ the real coordinate on $I$, we have that $d_I=dt*i(\partial/\partial t)$, where $i(\partial/\partial t)$ indicates the interior product with the vector field $\partial/\partial t$ and $*$ indicates the wedge product.
Why does the exterior derivative in $M$ split in that way? Then, the exterior derivative should $k$-forms into $k+1$-forms, but it doesn't seem to hold for $d_I$, if we define it like above.