Where is energy in energy density? I was learning about energy density and it seemed to be defined as the potential energy per unit volume in an electric field
$\frac{dU}{dV} = \frac{1}{2}\epsilon E^2$
But how can just the electric field have a potential energy on its own without presence of any charge? What is causing this energy to be present in an electric field?
 A: There are some reasons for believing that fields are real as well as being a mathematical convenience; and that they do contain energy.  A prime example would be the electromagnetic wave, which consists of fields that can carry energy from one point to another. Another example would be a magnet, which can push or pull another magnet. Where does the energy come from to do that work? The only thing that changes is the configuration of the fields.
A: 
But how can just the electric field have a potential energy on its own without presence of any charge? What is causing this energy to be present in an electric field?

Remember the mechanical definition of energy. Energy is the capacity to do work. The electric field can exert a force on a moving charge, so it does have energy. Furthermore, the field itself must be able to carry the energy since energy can leave all of the charges. Finally, the field energy also represents other definitions of energy, like the Noether theorem based definition.
A: In particle based electrical physics, we treat particles as having some charge, and they interact with each other.  This is a reasonable approximation for slow speed activity.  For higher speed activity (such as trying to understand how an ethernet cable or Wifi router transmits data), we have to use a more accurate model which captures more of the behaviors: an electric field.  The equations you are looking at more realistically handle propagating signals.  If you're used to just working with charged particles, this is the next step.  (wait until we tell you that light is a particle and a wave!)
Why fields?  Because they just work.  I have a few answers out there on Stack Exchange speaking to this, but it's not like physics chose to use fields.  We simply observed that, if we model the world with these fields, we get answers which are consistent with reality.  We use fields because they work, and no other reason.
In electric field theory, we deal with a field which is present at all locations in space and time, rather than just a few points.  And instead of summing the effects of these points, we integrate over the field (which is really just the continuous version of summing things up).  It just so happens that the electric field is  almost divergence free.  When you work through the math, you'll find that the divergence of the field is the charge at that point.  If you're tying together the particle based thinking and the field based thinking, this is where they tie together.  At every point where you have a charge, you have a divergence; at all other points, the field is divergence free.  In some special cases, your integrals become dependent only on the divergent parts of the field -- and out pops the particle based equations you are used to!
When you look at the math, you will also see that the field is conservative.  Because of this, we can define a function which is the "potential" of this field.  With a little more math, we can see that it is an energy term.
You have more experience with energy being bundled up without a local charge than you might think.  The electric field is not the only field out there.  There are many others, including magnetic fields.  There's a curious interplay between electric and magnetic fields that let them pass energy back and forth between them (they're coupled by Maxwell's equations).  It's like two dancers spinning across the dance floor.  If you analyzed either one, they're leaning so far back that they'll fall, but they hold onto eachother, and can twirl across the floor.  Likewise, the electric field and magnetic field can create little pockets of energy that swirl across the cosmos.  We call these interplays "light," and its how a charge that's millions of light years away can transmit light across space, dancing across the stars until it reaches the back of our retina, where the electric field portion of that dance interacts with electrons in our rods and cones, and we see it as light.
A: There are charges present. They are the cause for the electric field. Defining something based on the electric field is indirectly defining it based on the charges and charge configuration.
A: The expression for the energy density of the electromagnetic field follows in quite a straightforward way from the Maxwell equations, as what is known as Poynting theorem:
$$
-\frac{\partial u}{\partial t}=\nabla\cdot\mathbf{S} + \mathbf{J}\cdot\mathbf{E},
$$
where the energy density is given by
$$
u=\frac{1}{2}\left(\epsilon_0\mathbf{E}^2 +\frac{1}{\mu_0}\mathbf{B}^2\right).
$$
I stres sthat this derivation is general - it does not require any additional assumptions regarding charges and currents, and the fields in the expression above are the full electric and magnetic field.
In the electrostatic case the magnetic field is absent, i.e., the Poynting vector $\mathbf{S}$ is zero, and the current density is zero as well (since the charges are static), which leaves us with
$$
u=\frac{\epsilon_0\mathbf{E}^2}{2}
$$
