Complex Numbers, Phasor Notation, and Eulers Identity
I'll just zero in on the last bit.

The authors are using bold, capitalized V and I to indicate that they are complex values. Not merely real values. That's all it is.

So, assume \$\mathbf{V}=5 j\$. Then:

$$
\begin{align*}
V\left(t\right)&amp;=\operatorname{Re}\left(\mathbf{V}\cdot e^{j \omega t}\right) \\ &amp;= \operatorname{Re}\left(\mathbf{V}\cdot \left[\operatorname{cos}\left(\omega t\right)+i\operatorname{sin}\left(\omega t\right)\right]\right) \\ &amp;= \operatorname{Re}\left(5 i\cdot \left[\operatorname{cos}\left(\omega t\right)+i\operatorname{sin}\left(\omega t\right)\right]\right) \\ &amp;= \operatorname{Re}\left( 5 i\cdot\operatorname{cos}\left(\omega t\right)+5 i\cdot i\operatorname{sin}\left(\omega t\right)\right) \\ &amp;=\operatorname{Re}\left( 5 i\cdot\operatorname{cos}\left(\omega t\right)-5 \operatorname{sin}\left(\omega t\right)\right) \\ &amp;= -5 \operatorname{sin}\left(\omega t\right)
\end{align*}
$$

But perhaps \$\mathbf{V}=3-4 j\$? This just provides phase information where the polar magnitude is \$5\:\textrm{V}\$. But the result is arrived at in the same fashion as above. It's just that you now have sine and cosine terms mixed into the result.