Step 1: Start from charge instead of voltage.
The energy of a charged capacitor can also be written as \(U = \dfrac{Q^2}{2C}\), where \(Q\) is the charge on either plate.
Step 2: Insert parallel-plate quantities.
For a parallel plate capacitor, the surface charge density is \(\sigma = \dfrac{Q}{A}\), so \(Q = \sigma A\), and the capacitance is \(C = \dfrac{\varepsilon_0 A}{d}\).
Step 3: Substitute.
\(U = \dfrac{Q^2}{2C} = \dfrac{(\sigma A)^2}{2\left(\dfrac{\varepsilon_0 A}{d}\right)} = \dfrac{\sigma^2 A^2 d}{2\varepsilon_0 A} = \dfrac{\sigma^2 A d}{2\varepsilon_0}\).
Step 4: Divide by the volume \(Ad\).
\(u = \dfrac{U}{Ad} = \dfrac{\sigma^2}{2\varepsilon_0}\).
Step 5: Replace \(\sigma\) by the field.
The field between the plates is related to the surface charge density by \(E = \dfrac{\sigma}{\varepsilon_0}\), so \(\sigma = \varepsilon_0 E\). Substituting, \(u = \dfrac{(\varepsilon_0 E)^2}{2\varepsilon_0} = \dfrac{\varepsilon_0^2 E^2}{2\varepsilon_0} = \dfrac{1}{2}\varepsilon_0 E^2\). This is the energy stored per unit volume of the electric field, and it holds for any electric field, not just inside a capacitor.
\[\boxed{u = \dfrac{1}{2}\varepsilon_0 E^2}\]