According to this video here on the Casimir Effect Wikipedia. Due to the greater pressure from the 'outside' of the plates these plates of metals get pushed towards each other. When this pressure pushes one plate it does work on the plate and increases the plate's energy. However since the effect is demonstrated in a vacuum where does this energy come from?
However since the effect is demonstrated in a vacuum where does this energy come from?
You are under a bit of a misapprehension here. The many different fields described by Quantum Field Theory exist everywhere throughout the universe, they are not limited by a vacuum.
In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics. QFT treats particles as excited states of the underlying physical field, so these are called field quanta.
Although the Casimir effect can be expressed in terms of virtual particles interacting with the objects, it is best described and more easily calculated in terms of the zero-point energy of a quantized field in the intervening space between the objects. This force has been measured and is a striking example of an effect captured formally by second quantization.
Personally, my mental image of it is that there are less excitations/modes of vibration inside the plates than outside, so there is a net force. It is not that there is no field in the vacuum, it's that there are less degrees of freedom for the field inside the gap than outside it.
I apologise to all of those people now shouting at the screen, who have gone (much) further in their studies of QFT than I have. I would ask for a little leeway in this crude descrption, and hopefully a more sophisticated answer. There are probably duplicates and better answers currently on this site.
Think about how the plates get there in the first place. The energy of the vacuum state with the plates placed apart has higher energy than the vacuum state with the plates together (or with no plates at all). Therefore, putting the plates in their initial positions does work on the quantum fields comprising the vacuum. This work input can then be extracted again by releasing the plates and allowing them to be pushed together.
You can think of the placement of the plates as like a piston (plates) compressing a gas (quantum fields). Releasing the plates allows the vacuum to perform work on the plates, just like a compressed gas would perform work on a piston by expanding back to its equilibrium volume. Clearly you never get more energy out than you put in.