Just a quick preliminary answer, I will fix it later.
The connection to general relativity is a change of variables in which the metric is replaced by a "spin connection" and a "frame field". These quantities can then be arranged in a new matrix, so the metric field has been rewritten as a different matrix-valued field, and the transformations (diffeomorphisms) allowed under the symmetry of general relativity (general covariance) map to gauge transformations of this new matrix-valued field. The commutation relations above, are for the group of these gauge transformations - J corresponds to translations, P to rotations and boosts. The actual group is different depending on whether we are in flat space, de Sitter space, or anti de Sitter space; the cosmological constant (which is respectively zero, positive, negative) shows up in the commutation relations as lambda. d=3 is special because only there is a gauge-invariant action for this rewrite of general relativity possible. ISO(2,1) is just the special case of lambda=0, flat space in 2+1 dimensions.
All this is scattered through section 2 of Witten's paper. Also see part 1.1 of the sequel.
Thanks to T.S. for a discussion of this and related papers a few years ago.