These values are "SH-like" local support values:
Basically, FastTree does a very fast assessment of the support for each
split in the estimated ML tree by performing an SH-test on the three
nearest-neighbor topologies of each split. This is also done in PhyML,
and is discussed in section 1.6 of this paper:
Guindon, S., Delsuc, F., Dufayard, J.-F., & Gascuel, O. (2009).
Estimating maximum likelihood phylogenies with PhyML. Methods in
molecular biology (Clifton, N.J.), 537, 113–137.
doi:10.1007/978-1-59745-251-9_6
And here is the paper describing the SH-test:
Shimodaira, H., & Hasegawa, M. (1999). Multiple comparisons of
log-likelihoods with applications to phylogenetic inference.
Molecular Biololgy and Evolution, 16, 1114–1116.
These values are calculated by FastTree on a fixed alignment (i.e., the
alignment of a given iteration of of SATe) and thus do not capture any
uncertainty in the alignment. In other words, these values are not
summarizing any variation in the trees among iterations of the SATe
algorithm.
Like bootstrap support values (and unlike Bayesian posterior
probabilities), they are non-parametric.
I hope this helps,
Jamie