To understand this derivation you need to understand the concept of work done and have a good ability to rearrange and substitute into equations.

To derive most energy equations, you begin with the definition for work done:

$W = F x$    (1)

The difference between elastic potential energy compared with kinetic energy or gravitational potential energy is that when you start stretching an elastic band or a spring, it is really easy to extend it, but the more you stretch the harder it becomes and the more force you need to exert on it (assuming you never break or deform it).

You will have done an experiment at some point no doubt which compares the load that you apply to a spring and the amount it stretches by. The results are as follows:

This relationship is known as Hooke’s law and it tells us that the force is directly proportional to the extension of the spring;

$F \propto x$

The limit of proportionality is dependent on the type of spring and its stiffness, this quantity is known as the spring constant and is denoted with the letter $k$. We can therefore write:

$F = kx$    (2)

For a certain distance that the spring would extend (that we will call $x$), we can say that there is a maximum force required to hold it there, or that an average force was required over the entire distance (because a much smaller force is required to begin with and a higher force later on). We can substitute this idea of an average force into equation (1) along with the distance symbol $x$:

$W = \frac{F}{2}x$    (3)

Substituting equation (2) into equation (3) gives:

$W = \frac{kx}{2}x$

Which can be rearranged to give $W = \frac{1}{2}kx^{2}$

The extension of the spring frequently has the letter $e$ instead of $x$ and so the equation becomes:

\$latex W = \frac{1}{2}ke^{2}

Or in a similar way to how you are given it in your examinations:

$W = 0.5 k e^{2}$

Since this type of energy is elastic potential energy, the $W$ can just be changed into $E_{e}$, really you can use whatever symbol you want but to help an examiner mark your examination, use $E_{e}$ !

$E_{e} = 0.5 k x^{2}$3

This video helps explain this derivation too: