If we know the classical physics theory of the electromagnetic force (and we do), we can guess what the quantum mechanics theory for it should be (and then test with experiment, and as far as we can tell we've guessed correctly). We can do likewise with any classical force. (Although the strong and weak theories were not found by starting from any classical force but by generalizing the electromagnetic theory.)
These theories are extremely hard to calculate with exactly, but we can make a series of successively more accurate approximations. The first step works without hick-ups, but in the second step you run into things that are infinite or ill-defined. Some of them can be understood as for example an electron interacting with the electric field it itself produces.
Luckily there exists a method called renormalization by which we can interpret the theory in such a way that everything remains finite and unambiguous. However, it is not a matter of course that this method will work. The 1999 Nobel Prize was awarded for showing that it worked for the strong and weak theories. As mentioned renormalization is necessary in part because of self-interactions. It turns out that if we write down the quantum version of the classical theory of gravitation -- general relativity -- the self-interactions are such that we do not know how to apply renormalization.
So what is really meant is that we don't know how to construct and interpret a quantum field theory of gravitation so that it (1) reproduces the classical theory, and (2) does not spit out infinities when we try to calculate in more detail.
