Please explain the logic behind $d(xy) = y(dx) + x(dy)$

Hint

Suppose a rectangle of dimensions $x$ and $y$; its area is $A_0=xy$. Now change $x$ to $x+\Delta x$ and $y$ to $y+\Delta y$. The area of the new rectangle is given by $$ A_1=(x+\Delta x)(y+\Delta y)=xy+x \Delta y+y\Delta x+\Delta x \Delta y$$ So, the change of the area is $$\Delta A=A_1-A_0=x \Delta y+y\Delta x+\Delta x \Delta y$$ Now, let us make $\Delta x $ and $\Delta y$ very small; then $\Delta x \Delta y$ is negligible.

I am sure that you can take from here.

In order to illustrate, let us consider $x=10$ meters, $y=5$ meters and $\Delta x=\Delta y=1$ centimeter that is to say $0.01$ meters. So, using the last formula, $$\Delta A=10 \times 0.01+5 \times 0.01+0.01\times 0.01=0.1501$$ You see how small is the last term compared to the previous ones.