Quantifying the transport processes in galaxy clusters is an outstanding problem that has implications for our understanding of their thermodynamic history and structure. As the dilute plasma of the intracluster medium (ICM) is strongly magnetized, heat and momentum are transported preferentially along magnetic field lines. This anisotropy triggers a new class of instabilities that destabilize the ICM. We focus on the magneto-thermal instability (MTI), which is thought to be active in the periphery of galaxy clusters. Our aim is to take a fresh look at the problem and present a broad theory that (a) explains the MTI saturation mechanism and (b) provides scalings and estimates for the turbulent heat transport, in particular. We simulate MTI turbulence with a Boussinesq code, SNOOPY. The use of a Boussinesq model allows us to carry out an extensive sampling of the parameter space and to disentangle the effects of entropy and temperature stratification. We observe that the strength of entropy stratification sets an upper limit on the size of the turbulent eddies, and that g-modes are continuously excited at large scales by the MTI turbulence. Additionally, we note that, despite the tangled geometry of the magnetic field, a substantial fraction of heat is transported across the domain. In two dimensions, we find that the saturation regime involves a complex transfer of energy in spectral space, with energy being injected in the form of density fluctuations at small scale, and converted into kinetic fluctuations at large scales. In contrast, in three dimension this disparity of scales is absent and the turbulent motions drive a magnetic dynamo that is efficient at both high and low Pm. Finally, we validate our findings using a fully-compressible MHD code, PLUTO, and discuss potential applications of the instability to real clusters.