What is the geometric interpretation behind the method of exact differential equations?

Given an equation in the form $M(x)dx + N(y)dy = 0$ we test that the partial derivative of $M$ with respect to $y$ is equal to the partial derivative of $N$ with respect to $x$. If they are equal, then the equation is exact. What is the geometric interpretation of this?

Further more to solve the equation we may integrate $M(x) dx$ or $N(y)dy$, whichever we like better, and then add a constant as a function in terms of the constant variable and solve this.

e.g. If $f(x) = 3x^2$ then $F(x) = x^3 + g(y)$.

After we have our integral we set its partial differential with respect to the other variable our other given derivative and solve for $g(y)$. I have done the entire homework assignment correctly, but I have no clue why I am doing these steps. What is the geometric interpretation behind this method, and how does it work?

Great question. The idea is that $(M(x), N(y))$ defines a vector field, and the condition you're checking is equivalent (on $\mathbb{R}^2$) to the vector field being conservative, i.e. being the gradient of some scalar function $p$ called the potential. Common physical examples of conservative vector fields include gravitational and electric fields, where $p$ is the gravitational or electric potential.
The differential equation $M(x) \, dx + N(y) \, dy = 0$ is then equivalent to the condition that $p$ is a constant, and since this is not a differential equation it is a much easier condition to work with. The analogous one-variable statement is that $M(x) \, dx = 0$ is equivalent to $\int M(x) \, dx = \text{const}$. Geometrically, the solutions to $M(x) \, dx + N(y) \, dy = 0$ are therefore the level curves of the potential, which are always orthogonal to its gradient. The most well-known example of this is probably the diagram of the electric field and the level curves of the electrostatic potential around a dipole. This is one way to interpret the expression $M(x) \, dx + N(y) \, dy = 0$; it is precisely equivalent to the "dot product" of $(M(x), N(y)$ and $(dx, dy)$ being zero, where you should think of $(dx, dy)$ as being an infinitesimal displacement along a level curve.