What does the Hermiticity have to do with the conservation of energy?

Let's assume a Hamiltonian which is not time-dependent. If $H$ is Hermitian, then the time evolution operator $U(t)=mathrm e^{-mathrm i frac{t}{hbar}H}$ is unitary. So then the expected energy is constant:

$$begin{align}langle H(t)rangle&=langlepsi(t)mid Hmidpsi(t)rangle &=langlepsi(0)mid U^dagger(t)HU(t)midpsi(0)rangle &=langlepsi(0)mid U^dagger(t)U(t)Hmidpsi(0)rangle &=langle psi(0)mid Hmidpsi(0)rangle &=langle H(0)rangle end{align}$$

You need to use that the time evolution operator commutes with the Hamiltonian (because it's an exponential of the Hamiltonian) and that it's unitary. So at least one of these assumptions must be broken if we want to model a system without conservation of energy. Now, $U(t)$ must commute with $H$, unless we throw out the Schrödinger equation, too. So we have to make $U(t)$ non-unitary. But if $H$ is Hermitian, then $U(t)$ is automatically unitary, so $H$ can't be Hermitian for this to work. So the Hamiltonian of a system without conservation of energy must be non-Hermitian.

That they call such a Hamiltonian "effective" just means that if we disregard the outside influence which breaks conservation of energy, we can effectively describe the system using a non-Hermitian Hamiltonian. Of course, the true Hamiltonian of the entire system, outside influences included, is Hermitian, but we don't always care about those outside influences, and we can simply model them using a non-Hermitian Hamiltonian, and the results will be fine.