special relativity - Why do we identify symmetric 2nd rank tensors with spin-2 particles in string theory?

I am going through Tong's lecture notes on String Theory and came across the following irrep decomposition (Chap 2, p.43) of the bosonic string first excited states:

$$\text{traceless symmetric} \oplus \text{anti-symmetric} \oplus \underbrace{\text{singlet}}_{=\text{trace}}$$

He then goes on and claims that the traceless symmetric tensor is the spin-2 graviton.

What is the reason behind that claim? Is there a relationship between degrees of freedom and the spin of a particle in any number of dimensions? I remember from the $SU(2)$ irrep decomposition that the $\ell=1$ irrep has 3 d.o.f. just like a massive spin-1 particle would have. But what about massless particles living in 26 dimensions?