Disillusioned and skeptical like you, I hated that notation and the lack of meaning to it when I first learned about it in high school. For that reason, I decided to write 80 pages about it in my Bachelor's Thesis in the History of Mathematics. The historical subject of the Leibnizian Calculus is very deep and detailed, but here I will attempt to scratch the surface:
In some textbooks $\frac{dy}{dx}$ is used as notation for $f'(x)$. In that sense the middle step below
$$
\frac{dy}{dx}=\frac{g(x)}{h(y)}\iff h(y)dy=g(x)dx\iff\int h(y)dy=\int g(x)dx+c
$$
really does not make much sense in itself. If $dx$ and $dy$ were never introduced formally as true entities by themself this has to be a rule for memorization purposes only.
Historically, however, $dx$ and $dy$ were given meaning individually. For any variable $t$, Leibniz defined $dt$ to denote the infinitesimal increment in $t$ when it goes from one state to the next. In that way a variable $t$ underwent consecutive stages
$$
t,t+dt,(t+dt)+(dt+ddt)=t+2dt+ddt,etc.
$$
with $dt+ddt$ being the next state of the (infinitesimal) variable $dt$ and $ddt$ being infinitely smaller than $dt$. One could go on forever in that way. Leibniz then put forward the principle, that on each level of being finite, infinite or infinitesimally small, the so-called order of infinity, variables that differed only by a lesser order of infinity could be considered equal.
All this had close connection to geometrical interpreations - for instance lines pointing in directions differing by an infinitesimal angle were considered to be parallel. Also, if $ds$ denoted an infinitesimal segment of a curve in variables $x,y$, then $dx^2+dy^2=ds^2$, an infinitesimal version of the Pythagorean Theorem, so that $ds$ was considered a straight line segment rather than a curve segment in that setting.
As an example, given $y=x^2$ we have $(y+dy)=(x+dx)^2$ since the next stages for $x,y$ should also satisfy the equation. Subtracting consecutive stages we then have:
$$
dy=(y+dy)-y=(x+dx)^2-x^2=2x\ dx+dx^2
$$
and by Leibniz's principle $dx^2$ is infinitely smaller than the rest and can thus be disregarded.
If we should have done the same in a modern setup, we could write $y=x^2$ to have
$$
\frac{\Delta y}{\Delta x}=\frac{2x\Delta x+\Delta x^2}{\Delta x}=2x+\Delta x\rightarrow 2x
$$
and in that sense Leibniz's $2x\ dx+dx^2=2x\ dx$ corresponds exactly to the last step here were we go to the limit $\Delta x\rightarrow 0$. Note that $\Delta$ is used to denote actual finite differences greater than zero, whereas $d$ is used to denote infinitesimals.
I like to say that Leibniz worked permanently in the limit in some sense.
CAUTION
One should a bit catious here, though. The connection between Leibniz's calculus and modern calculus may not be as simple as the example of $y=x^2\implies dy=2x\ dx$ would suggest. Although we have the perfect connection $f'(x)=\frac{dy}{dx}$ this way, the next step
$$
f''(x)=\frac{ddy}{dx^2}
$$
is not always true. It is only true if we assume $ddx=0$ which requires a slightly deeper introduction to Leibniz's calculus to know what is meant by that. Also
$$
h(y)dy=g(x)dx\iff\int h(y)dy=\int g(x)dx+c
$$
is not trivially found to be the limit for some relation for finite differences:
$$
h(y)\Delta y=g(x)\Delta x\text{ and }\sum h(y)\Delta y=\sum g(x)\Delta x
$$
I once tried proving some connection for the above using the Mean Value Theorem, which worked out well but was not trivial.
As a final remark, note also that Leibniz's idea of a continuous variable increasing through distinct well defined stages has been problematized. The ontological status of infinitesimals was widely disputed in the centuries after Leibniz. A bishop, Berkeley, who was also versed in mathematics, was one of the early critics ridiculing and calling Newtons and Leibniz's infinitesimals Ghosts of Departed Quantities.
