I recall hearing flux described as the amount of rain passing through an open window. Depending on the direction of the rain and the orientation of the window, all of the rain, some of the rain, or none of the rain may pass through.
For example, consider a rainstorm where every raindrop is coming perfectly straight down, and you have an open window on your roof that is perpendicular to the direction of the rain. The flux would be exactly equal to the rate of water entering your home divided by the area of your window. If rain somehow stayed perfectly still after striking your floor, you would have a puddle of rain exactly the shape of your window on your floor.
Now, lets change the orientation of the window--say the same size window is now open on the side of an angled roof. If the rain is coming straight down, the magically immobile puddle on your floor would be smaller. It would be the projection of the shape of your window onto the plane perpendicular to the rain. The flux would be equal to the rate of water entering the home divided by the area of this projection.
Now, lets change the orientation of the window again -- say its now on one of your walls. Since the rain is coming straight down, no rain would enter your window since the opening is parallel to the rain. Since no rain is entering the home, the flux is zero.
The same analogy holds with more complicated rain patterns and window shapes. The trajectory of the raindrops is analogous to a vector field, and the window is analogous to any arbitrary surface on which you want to calculate the flux.