Why must the excitation of closed strings in String Theory be spin-2? In String Theory it is predicted that as a result of the closed strings we have spin-2 gravitons. 
1) How do we know there must be an excitation of spin-2 particles?
2) Why does a spin-2 particle HAVE to be a graviton?
 A: You ask: 
In String Theory it is predicted that as a result of the closed strings we have spin-2 gravitons.

1) How do we know there must be an excitation of spin-2 particles?

String theories have been chosen and are being extensively studied because they can have a representation of spin two particles . One searches for such theories because when attempting to quantize  gravitation the carrier particle, equivalent to the photon or gauge bosons, that appears in the quantum field theories studied, has spin two.
So only theories that can accommodate spin two particles can be candidates for a theory of everything.( strong, weak, electromagnetic and gravitational forces).
2) Why does a spin-2 particle HAVE to be a graviton?
It does not HAVE to, but if the theory under study does not have a spin two particle to be a candidate for describing the graviton, it cannot turn to be a Theory of Everything (TOE), which is the holy grail of theoretical physics.
A: The reason is that in string theory, every operator on the world sheet creates a physical particle or a superposition of particles. When you make a small change in the background metric that the string is moving along, this changes the action by a certain operator, which means that a background geometry is equivalent to some particle in the string theory. The coordinate invariance means that if you change the metric in a coordinate way, nothing happens, and this gives the gravitational ward identity. This is worked out in chapter 2 of Green Schwarz Witten, and it is an insight most straightforwardly explained by Tamiyaki Yoneya in the late 1970s and early 1980s.
The reason it is spin 2 is the same as in GR--- it's because it's a small change in a metric, and a metric is spin 2 (two symmetric indices).
A: have a look at http://en.wikipedia.org/wiki/Graviton. This explains that a spin 2 particle must be equivalent to general relativity. The details of why this is are far over my head I'm afraid.
