In the advection dominated case the problem can develop physics which is invisible to a computational mesh that is too coarse (say by having elements which contain many wavelengths). This leads to instability because neighboring elements could disagree considerably on what they believe are the right values to represent this physics (think interpolating high frequency waves with low order polynomials, you can get wildly different interpolants depending on how you sample.). This is problematic to continuous Galerkin because despite this potential disagreement, it forces continuity between elements. This is why you have mesh refinement restrictions, basically you only see stability once you have correctly resolved the advection (or add a bit of magic to the formulation).
DG resolves this by not requiring continuity and instead adding a term to penalize jump discontinuities, this penalty term essentially adds artificial dissipation to the method if the solution develops large jumps. This typically yields unconditionally stable methods (unconditional on the mesh size,polynomial order, and advection speed). Whether or not this is a desirable property depends on your application though, I think. Although the formulations are stable, it does not mean they are accurate. One could be stuck in the preasymptotic part of the convergence theory until they refine the mesh (resp. increase polynomial order), and then the level of mesh refinement used for convergence to kick in often ends up being exactly the same amount of refinement used for stability in the continuous cousin. Also be very careful about trusting stability results that depend on integrals being evaluated exactly.