Methylation of amines inside an introverted resorcinarene-based deep methyl ester cavitand is investigated by means of molecular dynamics simulations and quantum chemical calculations. Experimentally, the cavitand has been shown to bind a number of amines and accelerate the methylation reaction by more than four orders of magnitude for some of them. Eight different amines are considered in the present study, and the geometries and energies of their binding to the cavitand are first characterized and analyzed. Next, the methyl transfer reactions are investigated and the calculated barriers are found to be in generally good agreement with experimental results. In particular, the experimentally-observed rate acceleration in the cavitand as compared to the solution reaction is well reproduced by the calculations. The origins of this rate acceleration are analyzed by computational modifications made to the structure of the cavitand, and the role of the solvent is discussed.