homework and exercises - Photoelectric effect intensity

I understand the PE effect quite well but I'm failing to understand one thing. Intensity is the amount of energy per second incident to a given area.

So can you can increase the intensity by either increasing the electrical energy or making the area smaller? By increasing the electrical energy, what exactly is happening? I assume that by increasing intensity you are increasing the number of photons incident to an area of the metal thus increasing the rate of photoelectric emission. But why doesn't increasing intensity via more electrical power lead to greater energy of the electrons, if by increasing intensity you are increasing the energy of the incident waves?

I understand the results but I'm trying to understand why the intensity works the way it is thanks