Be careful: The energy dimension of the fields depends on the space-time dimension! You get the counting by looking at the kinetic terms of the field. E.g., for a scalar field, it's ##\partial_{\mu} \phi \partial^{\mu} \phi##. The action as a whole is dimensionless (when setting ##\hbar=1##, using natural units). ##\partial_{\mu}## has energy dimension 1 (or space-time dimension -1, which is equivalent when also ##c=1## is used). The space-time four-volume element ##\mathrm{d}^d x## has energy dimension ##-d## Thus the energy dimension of the scalar field must fulfill ##2d_{\phi}+2-d=0##, i.e., you have ##d_{\phi}=(d-2)/2##, which is of course 1 for ##d=4##.
Now think about the Dirac field. The kinetic term in the Lagrangian contains ##\overline{\psi} \gamma_{\mu} \psi##. So now you can easily do the dimensional analysis yourself.
Now think about ##\phi^4## theory. The interaction Lagrangian is ##\lambda \phi^4##. Now you can check, which dimension this coupling must have in ##d## space-time dimensions (only for ##d=4## the coupling is dimensionless). That explains, why it's necessary to introduce a energy scale into the theory in order to keep ##\lambda## dimensionless in any dimension. This is a rather indirect way, how the renormalization scale enters the very formal dimensional regularization, which has its merits through its convenience with regard to gauge symmetries, not as a tool to understand the physical meaning of regularization and renormalization, which is given by the Wilsonian point of view on renormalization. This is quite well treated in principle in Peskin & Schroeder. Unfortunately this otherwise very good book, ironically messes up the proper treatment of the renormalization scale, and (horribile dictu) you find dimensionful quantities as arguments of logarithms in the chapter on the Wilsonian treatment of renormalization, which is a real pity!).
