We present a fully three-dimensional theoretical study of the extraordinary transmission of light through subwavelength hole arrays in optically thick metal films. Good agreement is obtained with experimental data. An analytical minimal model is also developed, which conclusively shows that the enhancement of transmission is due to tunneling through surface plasmons formed on each metal-dielectric interface. Different regimes of tunneling (resonant through a "surface plasmon molecule," or sequential through two isolated surface plasmons) are found depending on the geometrical parameters defining the system.