Operational concept behind solving a general seperable form $\frac{dy}{dt}=f(t)g(y)$ by the integration $\int \frac{1}{g(y)} dy=\int f(t)dt$ Q. It is clear that while solving the differential equation $$\frac{dy}{dt}=t,$$the operator 'definite integration relative to $t$' makes an inversion of $\displaystyle \frac{d}{dt}$ to get $$\int \frac{dy}{dt} dt=\int dy=y=\int t dt+C,$$ with the aid of fundamental theorem of calculus. Also in the case of $$\frac{dy}{dt}=ty,$$ the chain rule guides us to say $$\frac{1}{y}\cdot \frac{dy}{dt}=\frac{d}{dt}(\ln|y(t)|),$$ and thereby $$\int \frac{1}{y} \frac{dy}{dt} dt=\int \frac{1}{y} dy=\ln|y|.$$But, what exactly is the reasoning behind solving a general seperable form $$\frac{dy}{dt}=f(t)g(y)$$ by the integration $$\int \frac{1}{g(y)} dy=\int f(t)dt.$$

As you have pointed out, separation of variables is premised on the chain rule, not just in the special example you have noted, but in general.
We have
$$\frac{dy}{dt}=f(t)g(y)\implies \frac{1}{g(y)}\frac{dy}{dt}=f(t).$$
Letting $G(\cdot)$ denote an antiderivative of $1/g(\cdot)$ and $F(\cdot)$ denote an antiderivative of $f(\cdot)$, the chain rule tells us we may write the above as
$$\frac{d}{dt}G(y)=\frac{d}{dt}F(t)\implies G(y)=F(t)+c$$
for arbitrary constant $c$.
