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.