As mentioned, this is not the first evidence for gravitational waves. The data from BICEP2 shows that there is a much higher amount of B-mode polarization than what is predicted by gravitational lensing alone. According to theory, this could only be due to higher amplitude tensor modes in the CMB than previously observed (or rather, lack of observed). These tensor modes are only caused by long wavelength gravitational waves that originated during inflation. Effectively, if it is right, this data rules out several alternative theories of inflation and theories on the nature of the early universe (it was my understanding that the Ekpyrotic model of the universe would be ruled out, but I may be mistaken so don't quote me on that)
So what this data points to is the first evidence for primordial gravitational waves. This agrees very well with elementary models of inflation and is only in slight tension with Planck data over the size of the slow-roll parameter for slow-roll inflation. This is not the first evidence for inflation. Rather, it is evidence that further agrees with inflation and rules out some of the non-inflation theories.
As far as I have seen, the results do not yet indicate that quantum gravity will be needed to explain them; however, I think there are some theories of quantum gravity that these result might support. Let me be clear because it was pointed out that what I said was misleading. Quantum gravity is the most promising method of explaining the results, but some models of inflation allow for other classical phenomena to generate primordial gravity waves. So while quantum gravity is preferred (especially among physicists that specialize in it), it is not yet required to explain it. That is, this is not solid evidence for quantum gravity since it does not rule out the possibility of classical mechanisms, but it can be taken optimistically as preliminary evidence. As for whether some other experiment will require quantum gravity to explain it, I can't speculate.