The instant way to see a problem is through the field equations of General Relativity if i tried to bypass formalism and other Quantum Field Theory ideas to offer a crude argument (better illustrated later by first studying Quantum Electrodynamic for example
http://en.wikipedia.org/wiki/Quantum_electrodynamics
where Special Relativity and QM work well together up to a point, but forced to move away from classical electromagnetism concepts in order to address questions that Quantum theory renders sensible ie you no longer deal with forces and point particles and perfectly known trajectories etc).
http://en.wikipedia.org/wiki/Einstein_field_equations
You try in them (field equations) to relate the geometry of spacetime (Riemann Tensor) with the energy momentum (stress) tensor. In other words the distribution of matter and energy defines the geometry.
http://en.wikipedia.org/wiki/Stress%...3energy_tensor
The source of the field ie particles are of quantum nature eventually yet this treatment insists on dealing with them as perfectly localizable sources that their position and momentum are perfectly known at the same time (eg see how GR solves the Kepler problem of Sun -Earth using a point source for Sun). The source is treated as having the kind of objective reality that the metric desires but the particles as quantum objects fail to behave in that manner. So instantly the field equations cannot be quantum mechanically consistent.
Try to write down a stress tensor for say an electron when we know its described by a wave function and can be only seen as a probability cloud. The metric geometry doesnt have a probabilistic nature but the sources do!
Read also other ways to visualize the problem here;
http://en.wikipedia.org/wiki/Planck_scale
From a Quantum field theory perspective Renormalization fails when you try to quantize gravity.
http://en.wikipedia.org/wiki/Renormalization
http://en.wikipedia.org/wiki/Third_quantization