The basic description of the mechanism behind Cepheid pulsations is given here:
The accepted explanation for the pulsation of Cepheids is called the Eddington valve, or κ-mechanism, where the Greek letter κ (kappa) denotes gas opacity. Helium is the gas thought to be most active in the process. Doubly ionized helium (helium whose atoms are missing two electrons) is more opaque than singly ionized helium. The more helium is heated, the more ionized it becomes. At the dimmest part of a Cepheid's cycle, the ionized gas in the outer layers of the star is opaque, and so is heated by the star's radiation, and due to the increased temperature, begins to expand. As it expands, it cools, and so becomes less ionized and therefore more transparent, allowing the radiation to escape. Then the expansion stops, and reverses due to the star's gravitational attraction. The process then repeats.
I can't really describe it much better than this, but let me know if it needs to be made more clear.
Note that this mechanism isn't restricted to helium II/III. Other classes of variable star operate in the same way. e.g. RR Lyraes and $\beta$ Cepheids.
The equilibrium state of stars
In all phases, stars are presumed to be in hydrostatic equilibrium: the outward force of pressure in the star (from the gas, radiation and sometimes electron or neutron degeneracy) is precisely balanced by the inward force of gravity. In many phases, we can additional regard the star as being in local thermodynamic equilibrium. What we really mean here is that the star is not generating (or absorbing) energy through expansion and contraction. This isn't true for, say, pre-main-sequence stars, which are only generating energy by their contraction toward the main sequence.
Cepheid pulsations are oscillations about the equilibrium state. They aren't linearly stable but the non-linear components mean that the oscillations aren't big enough to tear the star apart. The calculation of the frequencies is either a tricky sixth-order eigenvalue problem or a simulation of the oscillation from which the frequencies can be determined by analysing the output.
Which modes are excited is a similarly complicated problem. There are many things we haven't actually solved! But you can think of the system as a driven, damped oscillator. The driving is given by the opacity mechanism described above, and the damping is related to the equilibrium structure of the star. So, the excited modes are usually around where these match best, which may happen to be the fundamental mode (where the entire star expands or contracts as a whole) or the overtones (where different layers are expanding or contracting).