Change the random variable for calculating the expectation
The low of unconscious statistician (LOTUS) reads:
$E[u(Z)]=\int dz f_Z(z)u(z)$
By LOTUS on the right applied with $u(Z) \rightarrow h \circ g(X)$ you have $E[h(g(X))]$.
On the left instead always by LOTUS applied on $u(Z)\rightarrow h(Y)$ you have $E[h(Y)]$.
Since $Y=g(X)$ you are evaluating the same expectation written in two different ways and therefore the identity holds.