This is a concept I have never really understood. I mean to say is how can we include such a thing in a theory? How can we use them if we know that they don't actually exist? Are they some sort of calculation tool?
Resonance structures are a useful way of visualizing and representing the bonding in a molecule - they don't actually exist, but they are a useful tool to see what the bonding and structure of a compound is.
Bonding in a compound can sometimes be nicely represented using two-center, two-electron bonds - for example methane. The structure and reactivity of methane can be rationalised by representing it as a carbon with four hydrogens. Each "bond" can be represented by having two electrons, one from carbon, and one from hydrogen, in a bonding pair. You might draw this as a combination between an sp3 hybrid and a 1s orbital. In reality these hybrid orbitals don't exist either - they are merely a tool to simplify the bonding to localised bonding which is easier to understand. In reality there a multiple interactions between the carbons 2s and 2p orbitals, with the 1s orbital of the hydrogen. However, it is easier, and an accurate enough model to model methane with four localised bonds.
Resonanace is useful where this idea of two-center, two electron bonds breaks down - for example we know from experiment that the structure of benzene is a regular hexagon, with identical carbon-carbon bonds. However the idea of two-center two-electron bonds breaks down here - we can't assign a structure using the ideas we know - carbon makes four bonds, the shape of benzene is a regular hexagon with identical bonds. In this case we can represent benzene as an in-between structure between two structures that can be represented as nice two-center, two-electron bonds. When we draw benzene as a resonance structure of two rings with alternating double bonds, what we are trying to represent is that the real structure of benzene is somewhere between these two structures - it is neither of the structure, and it doesn't resonate - ie. flick really fast between the two structures - instead it is a mixture of the two structures. In benzene the structure we are trying to represent can be understood by considering each carbon to be sp2 hybridized - again these hybrids aren't real, but a tool to understand bonding more easily. Then we have the 6 p-orbitals in the plane of the molecule. In reality these combine two give 6 molecular orbitals. These give a final structure where the bonding is identical between all six carbon atoms, a kind of one and a half bond - but the shape and idea of this delocalised bonding is very important to the reactivity of benzene and other aromatics - which is where this idea of resonance starts to break down in some ways, but the structures can still be useful, but with a proper understanding of the bonding behind the structure.
How can we use them if we know that they don't actually exist?
Resonance contributors don't independently exist but rather a composite of the resonance contributors exists.
Are they some sort of calculation tool?
So no, resonance contributors don't exist. However, examination of resonance structures can lend insight into reactivity. Why might phenol's conjugate base be unusually stable? We can rationalize it by drawing multiple resonance contributors. These contributors are stabilizing because they show the delocalization of electron density from the oxygen to various sp2 ring carbons.
This, however, we cannot do for something such as cyclohexanol, which has no resonance contributors.