I know that the definition of something called a "field" is formally defined as the presence of a quantity at every point in space. In quantum field theory, does the excitation of a field mean that the value of the field is changing over time? What the heck is the meaning of "excitation"?!
The term excite in this context generally means something like to add energy to. So if we excite a field that means we add energy to it.
This can be used in either a classical or quantum context, though I'd guess it is most commonly used in quantum mechanics where it means changing a system from a lower to a higher energy quantum state. It's most commonly used for systems that have discrete states, e.g. exciting an electron in an atom, though it could be used for continuous states e.g. adding energy to an electron within a conduction band.
You need to be cautious about extrapolating this to quantum field theory. A classical field is generally an intuitively obvious object and adding energy to it is easily understandable. However a quantum field is an operator field not a physical object - a quantum field does not have energy and you cannot excite a quantum field. However you can excite the field in the sense that you add energy to modes described by the quantum field. Having said this, many of us routinely refer to exciting the quantum field and understand that is just a shorthand and not to be taken literally.