Correct me if I'm wrong, but the reason that quantum mechanics and general relativity are incompatible is because the quantum foam at Planck scales renders space-time discontinuous and doesn't allow Lorentz transformations to occur. See also this Phys.SE post. How does string theory solve this problem?
String theory does not say that GR or quantum field theory hold at those scales. It posits strings, and gets to the Planck scale and predicts what it might look like, foam or stringy things arising and changing and so on. At lower energies it is consistent with quantum field theory and GR. So, GR is a low energy description, and does not worry about the Planck scale. String theory also has a model for the graviton as some kind of string vibration or twist.
Unfortunately there are few exactly solvable models in string theory, and so a lot of the calculations are approximations and perturbations. Also there are multiple versions with a very large number of parameters so nobody knows which it exactly predicts. Also, some of the most reasonable models are supersymmetric, but supersymmetric particles have still not been detected and there is some concern that it might mean the lightest ones, which should exist, should have already been detected. This is true for supersymmetry also without string theory, though string theory may have versions where this non-detection is still ok. All these uncertainties and unresolved issues in string theory goes towards making it difficult for some people to take it as a theory. Still, it is still one of two or three (one is loop quantum gravity with its own problems) theories of everything (string and more generally M theory) or gravity
String theory gives some mathematical structure that resembles what we would like quantum gravity to appear as. The closed string has spin 2 fields that are a graviton for instance. The most recent major result is the AdS/CFT correspondence. This says that the interior of an anti-de Sitter spacetime has gravitation which corresponds to a conformal field on one dimension lower on the boundary. For the anti-de Sitter spacetime $AdS_5~\sim~SO(4,2)/SO(4,1)$ the boundary has $CFT_4$. With this is that local observables in the $AdS_5$ interior are difficult to describe according to variables on the boundary. This means that the more local an observable is in the $AdS_5$ interior the more nonlocal the corresponding observable is on the boundary. This has some curious prospects, for quantum field theory is formulated by local harmonic oscillators and the rest, and such a CFT on the boundary describes a completely nonlocal gravity in the interior.
We might expect that quantum gravity is nonlocal. Quantum gravity might be thought of as the propagation of a field, which is spacetime, on the same field. Consequently it should maybe not be that surprising that quantum gravity is nonlocal, and it is manifested in this manner. We have formulated quantum field theory according to an infinite number of oscillators on a spatial sheet that are local by virtual of commuting (Wightman conditions etc), and we should not be surprised to think that maybe we are "putting too much" onto nature this way. There is the field of noncommutative geometry, Connes et al, which is maybe a way to thinking about this differently.
We have not of course completely quantized gravity. At best all we have are some effective theories of quantum gravity that might apply in some limited manner, if at all. We might indeed find that gravity is really an emergent effect of quantum states and fields and is really not at all quantized in the way we have ordinarily thought. As yet string theory only gives some hints about quantum gravity, M-theory, the equivalency of string types, AdS/CFT correspondence.
You see, QM and GR are incompatible because of quantum foam, but string theory solves that problem. According to string theory the most fundamental components of nature are strings which have planck length, so fluctuations in quantum foam cannot exist because they do not affect strings which are the most fundamental components.