Why do we multiply the electric field by the area to get the flux when we should divide it by the area? I don't understand how they came up with the electric flux law, I think we should divide the electric field by the area instead of multiplying it.
 A: Mark's answer is a great way to think about flux. I do want to address the sort of "backwards" thinking present in the question though.
It is not like physicists said "I want to define electric flux. What should that definition be?". The quantity $EA$, or more generally $\int\mathbf E\cdot\text d\mathbf a$, turned out to be very useful in describing, understanding, etc. electric phenomena. Therefore, it was decided that this would be emphasized as the "electric flux".
Values are used in physics because they are useful in describing the world around us. So, it doesn't make sense to say "I think the flux should really be defined as the field divided by the area." Really a better question to ask would be "Is the quantity $E/A$ useful to describe something important about a physical system?"
A: Imagine you are standing by a river and you are holding a hoop (maybe a hula hoop, maybe a basketball rim, something circular). Now dip that hoop into the river facing upstream so that water can flow through it. How much water is flowing through the hoop? If you imagine the cylinder of water that passes through in $t$ seconds, then the volume is $V = vtA$, where $V$ is the volume of water, $v$ is the velocity of the river current (the height of the cylinder is $vt$), and $A$ is the area of the hoop ($\pi r^2$ where $r$ is the radius of the hoop). Now, imagine that the hoop becomes bigger. Will more or less water flow through?
Electric flux is like the river current. We want to measure how much total electric field is "flowing" through the surface. So, if the surface becomes bigger, we get more flux. We can even extend the metaphor. If you rotate the hoop so that it faces at an angle from straight upriver, then less water can flow through since the area facing the oncoming water is smaller. Similarly, if the electric field penetrates a surface at an angle, then there will be less flux. That's why the electric flux equation includes an angle component (or a dot product if you are studying electric fields as vectors).
