The term photoelectric effect is used to describe two different phenomena. The first is the one you are thinking of, i.e. the ejection of photoelectrons from a metal surface by visible/uv light, and the second is the ejection of electrons from atoms by X-rays (as used in X-ray photoelectron spectroscopy).
Ejection of photoelectrons by X-rays is basically ionisation. I'm not sure I'd restrict the term ionisation to mean only removal of the outermost electrons as your lecturer seems to.
Gases don't have a work function like a solid or liquid, because their atoms/molecules don't overlap strongly enough to form an electronic band structure. So there is nothing analogous to the visible/uv light photoelectric effect in metals. However the X-ray photoelectic effect is the same in gases as in solids and liquids. The energy of X-rays is so much higher than the bonding energy in solids and liquids that the photoelectron ejection is the same for all three phases.
Look at the review of Formaggio & Zeller, 2012, and in particular their eqn (14) for quasi-elastic neutrino scattering,
$$
\sigma \propto G_F^2 s ~,
$$
where $G_F\propto 1/M_W^2 ~.$
The cross-sections are always complicated, but when you are just exchanging a W (or Z) at very high energy (their Fig.2), with momentum transfer below the $M_W$ scale, the only player is the denominator in the W propagator, so Fermi's "prescient" $G_F$ in the amplitude; and so its square in the rate/cross section. From dimensional analysis, you then need a power of s, the c.m. energy square, to match the dimensions of energy, in natural $c=1=\hbar$ units. $\sigma$ has dimensions of inverse energy squared, so the power of s on the right hand side must be 1, as above.
In the lab/detector frame, for a target of mass m and negligible neutrino mass,
$s=(E_\nu +m)^2-p_{\nu}^2 \sim 2mE_\nu $, so the above cross section indeed goes linear in the neutrino energy.
Now, in the paradigm of the review, (14), the target is an electron, but, instead, you are interested in the quarks inside the nucleons of matter, and the events are inelastic as the nucleons disintegrate at slightly higher q, and the quark/parton distributions of quarks in nucleons further shape the answer. But, ultimately, nothing will efface the above brutal, primitive dimensional fact...
Almost nothing. Unitarity QFT constraints cannot allow this indefinite rise of the cross section with energy. And, indeed, at very high energies, (don't let me estimate them... above a TeV, the scale of SSB) something has to give. Indeed, it does: Higgs exchange effects come to the rescue and limit the rise of the cross section with energy. (This, estimated in 1973 by C.H Llewellyn-Smith provided a major salutary boost to the SM picture.)
Still, the rise with E as a power of about 1/3 is an empirical confluence of the struck relevant nucleon parton distribution effects averaged over momentum transfer, Ghandi et al 1996. I know of no easy argument to intuit it.
It is seen from mass attenuation vs energy plot thar compton scattering cross section goes through a maximum. It is low at low energy and high at high energy. Why? I want a non mathematical answer.
Best Answer
This is not a complete answer, but I think I partially understand why the cross-section falls off roughly like $1/E_\gamma$ at high energies. Note that the title of the question refers to probabilities, but the graph is of cross-sections. The probabilities are affected by competition from pair production.
High-energy behavior
In terms of relativistic transformation properties, we expect $\sigma=A/F$ to be invariant under longitudinal boosts, where $A$ is a Lorentz scalar and $F=mE_\gamma$ is the Moller flux factor . The only dimensionless Lorentz scalars we can form are $x=p_\gamma\cdot p_e/p_e\cdot p_e=E_\gamma/m$ and functions of $x$. The high-energy limit of the integrated Klein-Nishina formula is $\sigma\approx \pi r_0^2\ln x/x$, where $r_0$ is the classical electron radius, so $A\approx \pi/\ln x$.
So it seems that the high-energy behavior of the cross-section is mainly just kinematic, except for the rather gentle logarithmic dependence on energy that is present as well. Of course this is not really a fundamental answer, since I don't have any physical reason to offer as to why $A$ is only logarithmic in $x$ at high energies.
Low energies, free electrons
In the limit of low energies, the Compton scattering cross-section for a truly free electron is just the Thomson cross-section, which is classical and constant. I assume the Klein-Nishina cross-section falls off as you go down in energy from $mc^2$ and then approaches that limit. It would be interesting to understand why it falls off, which I don't understand. Mathematically, if you look at the integrated Klein-Nishina cross-section as a function of $x$, a bunch of terms cancel to first order for small $x$.
Low energies, in matter
In matter, it's not really clear to me whether it makes sense to talk about Compton scattering in a low-energy limit, and I don't know how this was defined in the graph the OP posted. Compton scattering is by definition scattering of a photon by a free electron. At very low energies, it will not make sense to describe any electrons (except maybe conduction electrons in a metal?) as free. I would think the cross-section would depend on the condensed matter physics. In a non-metal, below the threshold for the photoelectric effect, I would expect the cross-section for scattering to be zero.