GR is a field theory. The field is the metric tensor, which is a function of position in spacetime just like any other field. So just like any other field in principle it could be quantised to produce a quantum theory of gravity.
The trouble is that quantising a field is a mathematically tricky thing to do.
So suppose we have an EM field that is a function of position in spacetime, then it can be written as a sum of sines and cosines by Fourier transforming it. This is a 3D transform, and the "sines and cosines" are plane waves that can have any direction in space, but the basic principle is the same.
So QM tells us that the wavefunction for a free particle is a plane wave. The wavelength and frequency are related to the momentum and energy of the particle.
If the EM field can be written as an infinite sum of plane waves then is it possible that the EM field can be written as an infinite sum of free particles i.e. photons?
And the answer is that yes indeed the EM field can be written as an infinite sum of free particles i.e. photons.
So we quantise the EM field by writing it as a sum of the wavefunctions we get for free particles from the Schrodinger equation.
Photons aren't free particles. Photons interact with anything that is charged, e.g. electrons, and that means they aren't free. You have to include these interactions in your wavefunction, and that produces equations too hard to solve. In fact there is still some argument as to whether the mathematics of quantum field theory actually makes sense.
So we can only do calculations in QFT by using an approximation. Basically we start with the free field and introduce the interactions as perturbations of the free field.
Well each Feynman diagram is a representation of one term in the perturbations that we have to sum to actually do a calculation.
You have to calculate and add up an infinite number of Feynman diagrams to do the calculation, but luckily the infinite series converges pretty fast so in practice these calculations are fairly straightforward.
Basically this is taking the infinite sum you get from the first calculation and subtracting infinity from it to give a finite result.
Now this sounds pretty dodgy maths, and for a long time people were pretty suspicious of the renormalisation process.
However it has now been put on a sound mathematical footing, and it works. Using renormalisation we find we can do calculations in QFT and get answers that agree with experiment.
In fact QFT now works so well that no experiment anywhere has ever disagreed with calculations using QFT. Not at the LHC - not anywhere. It is a spectacularly successful theory.
