You start with some source of kinetic energy. This can be the wind, flowing water, or steam from burning coal. This kinetic energy is used to spin a turbine. An example would be the spinning blades you see on a windmill. This turbine is connected by a pole to a loop of wire, which also spins with the shaft. This loop is placed in between two magnets, so that it lies in a magnetic field. This picture illustrates the basic concept:
The magnetic field lines, shown in blue, pass through the loop at a certain angle. When the loop is horizontal, as shown in the picture, these lines don't pass through the loop at all. But, as the loop rotates, the field lines hit the loop at a steeper angle. The magnetic flux through the loop of wire has to do with the angle of the lines with respect to the loop.
It's like when you stick your hand out of a fast-moving car. When your palms face the ground, your hand slices through the air, but when your hand is straight-up, you feel a lot of air resistance. If your hand is the loop of wire, and the magnetic field is the air, then the magnetic flux is the amount of resistance you feel from the air.
As explained by Faraday's Law of Induction, when the magnetic flux in a loop changes, a voltage is generated in that loop. As the loop rotates with the shaft, its angle with the field lines changes, and therefore its magnetic flux changes. This "induces" or generates a voltage, which creates electricity. The faster the loop spins, the faster its flux changes, and a stronger voltage is generated. This is why, in the case of wind-power, stronger wind will lead to more electricity.
It is worth noting that the rate of change of the flux is not constant, and depends on the angle of the loop, and therefore the generated voltage is not constant, but rather "alternating". This alternating voltage produces an"alternating current", or AC , which is the type of electricity you'll find in your wall outlet.
Hope this helps!