Why are gravitons present among the modes of oscillation of the 'strings' in String Theory?
In this fourth Lecture of his string course , Lenny Susskind explains in a slightly technical and very accessible manner, why there is a spin 2 excitation for closed strings which can be interpreted a graviton.
To explain this, he writes down all the lowest possible excitations of a closed string.There can be left and right moving waves of different frequencies. Taking into account the so called level matching condition, which says that the energy of the right moving waves must match the energy of the left moving waves, all possibilities but the ones with spin 2 and spin 0 are excluded. Physically, the level matching corresponds to the fact that the translation of the wave function along the closed string is a symmetry transformation. It means that when treating a closed string as a sequence of mass points, there is no special point on the string (they are all equivalent).
As I understand it so far, the interpretation of the spin 2 excitation as the graviton can be motivated by the following considerations: When discribing gravity as a QFT in a curved spacetime, the metric tensor takes the role of the gauge field. Because it has two Lorentz indices (the vector potential in QED has only one) this gauge field (the graviton) must be spin 2. In addition, in an interacting string theory, closed strings can not be avoided. If the coupling constant does not vanish, the ends of open strings can always come together and join such that the strings get closed. The revers process can happen too (with the same coupling constant), such the all (open and closed) strings can absorb and emmit closed strings or gravitons respectively. This corresponds to the fact that everything gravitates.
... so if something wags its tail like a dog, barks like a dog, etc then it probably IS a dog :-)