Why does the (relativistic) mass of an object increase when its speed approaches that of light?

The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to read this popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".

What actually changes at relativistic speeds is the dynamical law that relates momentum and energy depend with the velocity (which was already written). Let me put it this way: trying to ascribe the modification of the dynamical law to a changing mass is the same as trying to explain non-Euclidean geometry by redefining $\pi$!

Why this law changes is the correct question, and it is discussed in the answers here.

There is a point of view, that under the term "the mass" one must mean "the rest mass".

From that point of view there is obviously no dependence of the (rest) mass on the speed of an object. And, therefore, the mass of an object does not increase when its speed increases.

The correct (from that point of view) way to talk about the phenomenon is to say that with increase of the speed of an object you need more and more energy in order to make it move faster.

Of course there is no fundamental controversy between this point of view and that of many books and articles. But the usage of the concept of "relativistic mass" makes things much more complicated, even if it was introduced in pursuit of simplicity.

The complete relevant text in the book is

The de Broglie wave equation relates the velocity of the electron with its wavelength, $\lambda = h/mv$ ... However, the equation breaks down when the electron velocity approaches the speed of light as mass increases. ...

Actually, the de Broglie wavelength should be $$ \lambda = \frac hp, $$ where $p$ is the momentum. While $p = mv$ in classical mechanics, in special relativity the actual relation is $$ \mathbf p = \gamma m \mathbf v = \frac{m\mathbf v}{\sqrt{1-\frac{v^2}{c^2}}} $$ where $m$ is the rest mass. If we still need to make the equation $p = mv$ correct, we introduce the concept of "relativistic mass" $M = \gamma m$ which increases with $v$.