Chemical potentials of source and drain

It's a fancy way of saying that they hooked up a battery (or voltage source) across the device, and thus the source and drain are at different voltages. Note that they're refering to the electrochemical potential, which is not quite the same thing as the chemical potential (altho the terms are sometimes used interchangeably). In particular, the electrochemical potential is not constant in space. Per Ashcroft and Mermin Eq. 29.7, the electrochemical potential

$$mu_eleft(vec{x}right) = mu + e phileft(vec{x}right),$$

where $mu$ is the chemical potential and $phileft(vec{x}right)$ is the electric potential (i.e. the voltage due to the battery).

A better reference for your purposes might be the first chapter of "Quantum Transport" by Supriyo Datta, altho any book on transport or semiconductor devices will cover it. The topic is often briefly covered in solid state physics books --- usually in the context of a simple semiconductor device like a diode (as in Ashcroft and Mermin).

PS. The reference you cite is a very good one. In particular, it's precise with terminology. A lot of the time, the electrochemical potential will just be called the Fermi level, which I think is a confusing term. Also, as Vadim notes, the electrochemical potential in the island is actually a much thornier issue than in the source and drain --- large reservoirs, which can be treated as near equlibrium and thus have a well-defined chemical potential.