The dipole in the microwave background indicates motion of the Milky Way and thus of the whole Local Group, at least, at about 600 km/s in a certain direction. The straightforward explanation is that the density irregularities nearby from superclusters and voids result in a net gravitational acceleration that, over the age of the universe, resulted in this velocity. A number of superclusters (Hydra-Centaurus, Shapley, and Norma) happen to line up along this direction and reasonably fit the picture. Also there are some big voids in the opposite direction.
There are two basic ways to quantitatively understand what is the cause. One can make measurement of local flows around these superclusters and voids to determine their masses and thus their net effect on the MW velocity. Or one can try to measure the "dipole motion" on larger scales, that is at what size scale does this motion drop to 300 km/s and what scale does it drop to near 0? A number of groups have tried to do this and the results are in disagreement. In particular, one group found no drop off in motion out to fairly large distances. Therefore, an alternate explanation was proposed that the dipole motion is primordial and due to objects beyond the horizon that interacted with us before inflation.
Will we ever settle this? I am sure that we will eventually. There is much work to do measuring the distances of galaxies out to 100 Mpc or so with greater and greater accuracy using new methods (SN Type 1a, surface brightness fluctuations, Tip of the Red Giant Branch) and larger telescopes with better instrumentation and then modeling the peculiar flow on all scales. If the local distribution of matter cannot explain fully the dipole moment in the MWB, then perhaps Dark Flow from before inflation will be necessary to explain it.
