We report B-11 NMR studies that address the underlying two superconducting (SC) gaps model and a possible mechanism for interpreting a high SC transition temperature (T-c) of similar to40 K in a recently discovered MgB2. We reported in a previous paper [Phys. Rev. Lett. 87, 127001 (2001)] that the results of the nuclear-spin-lattice relaxation rate, 1/T-1 in the SC state revealed the existence of a single large SC gap with 2Delta/k(B)T(c)similar to5. This conclusion is corroborated by the calculation that any two-gap models cannot account for the experimental results. In Al-doped MgB2, the result of 1/T-1 in the SC state revealed that a size in the SC gap is not changed by substituting Al for Mg. The reduction in T-c by Al doping is shown to be due to the decrease of the density of states N(E-F) at the Fermi level. According to the McMillan equation, an experimental relation between T-c and the relative change in N(E-F) allowed us to estimate a characteristic phonon frequency omegasimilar to700 K and an electron-phonon coupling constant lambdasimilar to0.87. These results suggest that the high-T-c superconductivity in MgB2 is mediated by the strong electron-phonon coupling with high-frequency phonons.