Theoretically, we are supposed to believe that "Graviton" is the exchange particle for the gravitational interaction which is a spin 2 particle (Boson). What is the physical meaning of Spin 2 ?
The spin is the intrinsic angular momentum of a body. A spin-2 particle behaves under rotation like a playing card or needle that looks the same (is again in the same state) after half a rotation (180°).
Since others have already covered very well all the fundamental stuff, but as I see that you wanted to know specifically about the physical significance of the spin 2 nature of the graviton, I would like to mention something specific:
You may be aware that astrophysical events such as the inspiral and merger of binary black holes produce a general relativistic phenomenon called gravitational waves. It happens that for every orbit of the binary system, two gravitational waves are produced (thus, the frequency of the gravitational waves is twice the orbital frequency of the binary system), and this is directly linked to the graviton being a spin-2 particle. But, although gravitational waves have been directly detected, the graviton itself has not yet been.