Why does energy have to be emitted in quanta?
The book is almost surely referring to the ultraviolet catastrophe.
Classical physics predicts that the spectral energy density $u (\nu,T)$ of a black body at thermal equilibrium follows the Rayleigh-Jeans law:
$$u(\nu,T) \propto \nu^2 T$$
where $\nu$ is frequency and $T$ is temperature.
This is clearly a problem, since $u$ diverges as $\nu \to \infty$ (1). The problem was solved when Max Planck made the hypothesis that light can be emitted or absorbed only in discrete "packets", called quanta.
The correct frequency dependence is given by Planck's law:
$$u(\nu,T) \propto \frac{\nu^3}{\exp\left(\frac{h \nu}{k T}\right)-1}$$
You can verify that the low-frequency ($\nu \to 0$) approximation of Planck's law is the Rayleigh-Jeans law.
(1) To be more specific: if you consider electromagnetic radiation in a cubical cavity of edge $L$, you will see that all the frequencies in the form
$$\nu =\frac{c}{2 L} \sqrt{(n_x^2+n_y^2+n_z^2)}$$
with $n_x,n_y,n_z$ integers, are allowed.
This basically means that we can consider frequencies as high as we want to, which is a problem, since we have seen that when the frequency goes to infinity the energy density diverges. So, if we used the Rayleigh-Jeans law, we would end up by concluding that a cubic box containing electromagnetic radiation has "infinite" energy.
It is maybe this that your book is referring to when it says that "the whole energy in the universe would be converted into high frequency waves" (even if, if this is a literal quote, the wording is quite poor).
I think the author is referring to the UV catastrophy, a historical problem in physics which first led physicists to discover that electromagnetic energy was quantized.
The problem basically is this: For a system which is in thermal equilibrium each object is radiating and absorbing energy. Since it's in equilibrium the radiative energy emitted by any object in the system is equal to the energy absorbed by the rest of the objects. And the temperature of all the objects are equal.
When trying to calculate the distribution of this radiative energy among the EM spectrum, physicists found that theoretically the proportion of energy contained by radiation of frequency $\nu$ should be proportional to $\nu^2$! (see Rayleigh-Jeans law). This meant that as you went to higher frequencies the energy contained by them would go on increasing without limit so not only practically all the energy would be contained by higher frequencies, but also any system in equilibrium would have infinite energy. This is obviously not what we observe in real life so something was wrong.
It is only when they assumed that energy was quantized that they got a distribution law which not only made sense, but also fit the experimental data beautifully (see Planck's law).
I believe that this explains the context of the statement that the author was making but answering why does energy have to be quantized in reality is a deep, rather philosophical question to which no one really knows the answer.
Without quanta, an electron attracted to the nucleus of an atom would be accelerated as it orbits around it. By accelerated, I don't mean the layman meaning of speeding up but its velocity direction change. Now classical electromagnetism tells us that an accelerated charge emits electromagnetic waves. This is how an antenna produces radio waves for example by accelerating the electrons inside the antenna (in that case speeding them up and slowing them down in turn).
But then the energy radiated as electromagnetic waves implies that the electron looses energy to conserve the total energy, so basically the classical picture (that means without quanta) predicts an atom cannot be stable. So in that picture, all matter would nearly instantly collapse, leaving only a bath of electromagnetic waves.
That is one possible answer to your question. See also my fellows guess that it could refer to the black body problem!