The textbook writer is referring to the concept of relativistic mass, which is the idea that accelerating a body tends to become harder and harder as its speed approaches the speed of light. This is sometimes thought of in terms of an increase in the object's mass as the speed increases.
However, you should think of this as a deprecated concept that most modern physicists consider unnecessary and misleading. Nowadays, we prefer to think that the force-acceleration momentum needs to be rephrased to something along the lines of
$$
\mathbf F=\frac{\mathrm d}{\mathrm dt}\left[\frac{m_0\mathbf v}{\sqrt{1-v^2/c^2}}\right],
$$
with an invariant mass $m_0$, than to introduce a variable "relativistic mass" $m_R=m_0/\sqrt{1-v^2/c^2}$ in an attempt to clean up that relationship and make it look more classical.