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Why does a point charge generate an electric field and why is an electric field represented by this formula

$$E ~=~ \frac{q}{4 \pi \epsilon r^2},$$

where $\epsilon$ is permittivity of free space and $r$ is the distance from point charge $q$? Can this equation somehow be mathematically derived?

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Do you know Maxwell's equations? – Fabian Aug 7 '12 at 8:04
There is a point at which "Why?" questions have to be answered with "Because that is the way the world works.", and you've pretty close to it. Some things are simply experimental facts. We like to explain some of these facts in terms of fewer and simpler facts when we can but the basis is always agreement with the observed behavior of the universe. – dmckee Aug 7 '12 at 14:37
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There is a line of arguments which you can call a derivation. It is a fluid analogy which is called Gauss law.

I guess you are wondering why its inverse square law?

Its because of the sphere. Imagine you have the candle in the room. The far away you go from the candle the less light you have per unit area. It is so because the amount of light produced by candle is fixed but the area of the sphere on which it falls grows proportional to the square of distance from the candle. Because area of the sphere's surface is proportional to its radius squared.

Now imagine that electrical charge is also a candle which emits a special electrical rays. Amount of electrical "light" produced by the charge is fixed but the area on which it acts grows with the distance as radius squared so amount of "electrical" light per unit area will fall of like inverse square, and the force will reduce also like inverse square because we agreed it is proportional to the "brightness" of our "electrical" light.

Q is simply the strength of our "electrical" candle.

Other factors are experiment related and are not important for physics, except maybe for the Pi which is related to the area calculation.

You may want to investigate further into the Gaws law, which looks at electrical field as a flowing liquid. In case of one charge these two analogies are same, but in case of many charges Gaws law is more fundamental.

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