What does the derivative of area with respect to length signify? Suppose that we have a square sheet of edge length $L$. Its area $A=L^2$.  
Differentiating $A$ w.r.t. L, we get
$$\dfrac{dA}{dL}=2L$$  

I do understand what it means to differentiate, graphically, it gives you the slope of the tangent at a point on the graph. But now, when I think of what differentiating means in the context of Area and length, it doesn't make any sense to me at all. What does $2L$ signify? 

Try to draw a square $ABCD$ with side equal to $L$. Now draw a slightly bigger square $AB'C'D'$ with side length $L+\Delta L$ (such that $DD'=BB'=\Delta L$). Now look at the $\Gamma$-like shape cut from $AB'C'D'$ by $ABCD$, you can split it into three parts: two thin rectangles $L\times \Delta L $ and one small square $\Delta L\times \Delta L $.
Now the derivative is in quite simpified terms "the difference of value of the function over the change of argument", so basically when you increase the side length by $\Delta L$, then the surface increases by $2L\Delta L$ and a negligeble term $(\Delta L)^2 $.
One can also say that $2L$ signifies the permeter of the part of the square that got inflated.
