Why do spontaneous processes increase entropy?
First of all, take a look at this picture:
A reaction is said to be spontaneous if it occurs without being driven by some outside force.
Since your question is about entropy I'm continuing with it.
Entropy is a measure of the disorder of a system, and systems tend to favor a more disordered system (remember this!). Nature tends toward chaos. Funny, isn't it. Spontaneous reactions occur without outside intervention (force).
Back to the picture: when you mix two things (solute and solvent) you always get a solution. You can't get a solution where you have solvent on one side and solute on the other. It also tends toward chaos, right?
To conclude: spontaneous processes increase entropy!