In the fractional quantum Hall effect the electrons are effectively confined to move in two spatial dimensions. This means that whatever excitations comes out forth out of the collective behaviour of the electrons is also confined to these two dimensions. It's these excitations which can be viewed as quasiparticles living in a 2+1 dimensional world.
Your correct that the true world is still three dimensional. But the anyons in the fractional quantum Hall effect are described using an effective field theory. This effective field theory has a minimal length scale. As long as we do not approach this limit, we can employ the effective description. At some point the effective theory will break down, but then we lose whole the meaning of a quasiparticle anyway.
In the end, the funky statistics of the anyons is a reflection of the very complicated collective behaviour of the electrons. From some point of view you could also say that you are not even talking about statistics of the anyons, but rather a Berry curvature associated with the position of the quasiparticles. Braiding two quasiparticles means we are 'simply' adiabatically changing the wavefunction of the electrons instead. And it's this adiabatic change which induces a Berry phase (which is precisely the phase you expect if you just look at the statistics of the anyons).