Conditions for analytical signal reconstruction with the Hilbert Transform (HT)
Let me use the notation from Bedrosian's research report "The analytic signal representation of modulated waveforms" in a slightly simplified version:

$$s(t)=c(t)m(t)\tag{1}$$

where $s(t)$ is the modulated signal, $c(t)$ is a (not necessarily sinusoidal) carrier function, and $m(t)$ is the message signal. The carrier is assumed to be a relatively narrow-band band pass signal, and the message signal has a low pass character.

Let $c_a(t)$ be the analytic signal corresponding to $c(t)$. By definition, $c_a(t)$ has only positive frequency components. The analytic signal $s_a(t)$ corresponding to $s(t)$ can then be written as

$$s_a(t)=c_a(t)m(t)\tag{2}$$

if and only if the spectra of $c(t)$ of $m(t)$ do not overlap. Because only then does $s_a(t)$ as defined in $(2)$ contain positive frequency components only. If the spectra overlap, then $s_a(t)$ is not analytic because it will have negative frequency components. This is illustrated by the figure below, taken from Bedrosian's report:



The condition that the spectra of $m(t)$ and $c(t)$ do not overlap is equivalent to the requirement that $s(t)$ be a true band pass signal, i.e., $s(t)$ has no DC component.

I'm sure that this is what is alluded to by the sentence you quoted. I do agree that this is not obvious from the chosen formulation.