Gravity is non-linear, so if it is mediated by gravitons, gravitons must interact with each other. On the other hand, the effects of gravity moves with the speed of light, so if it is mediated by gravitons, gravitons must be massless.
How can gravitons interact with gravitons if they are not charged under gravity?
Gravitons couple to energy-momentum tensor $T_{\mu \nu}$, not mass (which captures only the $00$ component of $T_{\mu \nu}$). That's why photons (which are also massless, like gravitons) interact with gravitons leading to bending of light (Gravitational lensing).
The idea is that a free graviton field has its own $T_{\mu \nu}$, which must be added to the matter source's $T_{\mu \nu}$ to ensure that the conservation equation/Bianchi identity holds. But note that this free graviton's $T_{\mu \nu}$ is quadratic in the graviton field (say $h_{\mu \nu}$). This $O(h^2)$ term in the equations of motion can only come from an $O(h^3)$ term in the Lagrangian. This means that we have to update our free graviton theory to one that has a cubic interaction. But then this $O(h^3)$ term in the Lagrangian produces an $O(h^3)$ energy-momentum tensor, which must again be added to the previously-obtained sum of energy-momentum tensors in the equations of motion. This $O(h^3)$ term in the equations of motion can only come from an $O(h^4)$ term in the Lagrangian. This leads to a quartic interaction for the graviton field. And so on, until we sum up the entire series to obtain non-linear Einstein's theory with $n$-point graviton-graviton interactions for every $n$ from $n = 3$ until infinity. This was discussed several decades ago, and is nicely summarized in Deser's 1969 paper.
Gravitons are charged under gravity. They don't have mass, but they do have energy and momentum. And in general relativity, energy and momentum gravitate just as mass does. So gravitons gravitate (curve spacetime), just like more conventional matter does.
