If there is more overlap of the same sign wave function, this apparently leads to lower energy. I don't understand why. My professor said that there would be fewer nodes, and fewer nodes means lower energy. However, why would overlap in general cause the energy to decrease?
Although the nature of orbital and wave functions can be puzzling, by the end of the day chemical bonds are hold together by the familiar electrostatic forces. Although the electrons don't have a well-defined position, the forces felt by the nuclei is dictated by the spacial probability distribution of the electron.
To form a covalent bond, electron density has to accumulate between the two atoms. The energy is lowered because the electron is close to both nuclei. In a chemical bond such electron-nuclear attraction cancels out the repulsion between nuclei and holds the atoms together.
Orbital overlap changes electron distributions. Orbital with different signs cancel each other in and lower the relative probability electron appear between the bonding atoms. In the most extreme case of nodal plane, electron density is completely annihilated. Because the total number of electron is unchanged during interaction, the lowering of electron density between nuclei is accompanied by an increase of density to the far side.
In other words, when you have orbital with canceling phases interacting, electron density will move away from the bonding area, resulting in higher energy.
On the other hand, a strengthening interaction increase the electron density between nuclei, thus lowering the energy and creating a pulling force between nuclei.