Here is the simple explanation (credits should go to the late great V.I. Arnold who gave the explanation below in one of his books). The form $\omega$ is sometimemes referred to as the Gelfand-Leray residue.
Place yourself in the case when $t$ is a regular value of $g$. Fix a point p on the fiber $g^{-1}(t)$. at that point $dg(p)\neq 0$. The implicit function theorem then shows that we can find local coornidates $y^1,\dotsc, y^n$ near $p$ such that, in these coordinates, $g=y^1$. In these coordinates
$$dx^1\wedge \cdots \wedge dx^n=\rho dy^1 \wedge \cdots \wedge dy^n,$$
$$ \omega = \rho dy^2 \wedge \cdots \wedge dy^n. $$
Using these coordinates, and $a$ is supported in the domain of the coordinates $y^j$ we deduce from the Fubini theorem that
$$\int u a dx^1\wedge \cdots \wedge dy^n = \int_{\mathbb{R}} H_{y^1}[au] dy_1, $$
$$H_{y^1}[au]=\int au \rho dy^2\wedge dy^n= \int au \omega. $$
The relationship with currents is symple. The $n$-form $ udx^1\wedge \cdots \wedge dx^n$ defines a $0$-current. The $1$-form $H_{t}[u] dt$ is its pushforward.