In the introduction of the paper it is stated:
The recent determination of the proton radius using the measurement of the Lamb shift in the muonic hydrogen atom [1, 2] startled the physics world. The obtained value of 0.84087(39) fm differs by about 4% or 7 standard deviations from the CODATA  value of 0.8775(51) fm. The latter is composed from the electronic hydrogenate atom value of 0.8758(77) fm and from a similar value with larger uncertainties determined by electron scattering 
What Randolf Pohl et al basically did, is that they measured the radius of the proton using more accurate techniques. While they were expecting to just get a more accurate result(few more decimal places), their proton radius is clearly smaller than all other experiments before.
How they did the experiment is described in the paper I pointed to. The conclusion of the paper is:
The proton radius puzzle is real. A physical parameter should not depend on the method of extraction. Yet here we have two highly precise methods based on electronic and muonic hydrogen atoms that disagree significantly. The possible origins of the disagreement are the subject of much of this text. To summarize: the possibilities are
- The electronic hydrogen experiments are almost, but not quite, as accurate as stated (Sect. 2.2).
- The QED calculations are almost, but not quite, as accurate as stated (Sect. 4.2).
- The two photon exchange term that depends on proton polarizability has not been correctly evaluated (Sect. 4.3).
- The electron and muon really do have different interactions with the proton (Sect. 4.4), so that there is physics beyond the Standard Model.
None of these possibilities seem very likely, but all must be pursued.
My question is, what are the possible implications of this new result? Does anyone know how significant are the potential flaws in the experiments mentioned? It is stated in the paper that one possible solution might be Beyond the Standard Model theories; assuming the experiment is correct, do we really need BSM theories to explain the new results? If so, which specific branches of BSM will benefit from these results?