From the paper: "To account for the search background noise varying across the target signal space, candidate and background events are divided into three search classes based on template length. The right panel of Fig. 4 shows the background for the search class of GW150914. The GW150914 detection- statistic value of ρˆ_c = 23.6 is larger than any background event, so only an upper bound can be placed on its false alarm rate. Across the three search classes this bound is 1 in 203 000 years. This translates to a false alarm probability < 2 × 10^−7, corresponding to 5.1σ. A second, independent matched-filter analysis that uses a different method for estimating the significance of its events [85,86], also detected GW150914 with identical signal parameters and consistent significance" (https://dcc.ligo.org/LIGO-P150914/public). Take a look at Figure 4 as well.

In case you'd like to dig deeper, the 85 and 86 mentioned are:

[85] K. Cannon et al., Astrophys. J. 748, 136 (2012).

[86] S. Privitera, S. R. P. Mohapatra, P. Ajith, K. Cannon, N. Fotopoulos, M. A. Frei, C. Hanna, A. J. Weinstein, and J. T. Whelan, Phys. Rev. D 89, 024003 (2014),

It's not a counting experiment, which makes the calculation of a false positive rate somewhat harder. The key for LIGO is certainly that they saw the signal coincident at two stations, far apart.

This is about detection.

To put it another way, you need a single black swan to prove that black swans exists (to whatever sigma).

The detection uncertainty is a separate matter from the predicted rate. Sure, if you had a strong prior that GWs should be detected once in a billion years, then you would want a better detection. But as it is the priors on the detection are pretty weak and this is totally consistent with what is expected.

Both detectors detected the same event at slightly different times

Others have answered other aspects of this, but as I understand it, it is not the case that we don't know how rare they (BH-BH events) are because they are so rare, we don't know how rare they are, because we don't have a really good model for them. So, we don't know how often we'd expect to detect them, once we had a detector.