How to prove ${\bf u}\cdot{\bf v}=|{\bf u}|\cdot|{\bf v}|\cos\theta$, if $\theta$ is the angle between $|{\bf u}|$ and $|{\bf v}|$ This is a snippet from my book.

How did they get from $|{\bf u}|^2={\bf u}\cdot{\bf v}=|{\bf u}||{\bf v}|\frac{|{\bf u}|}{|{\bf v}|}$?

I dont understand the comments from your book, but I assume the question is how to prove the equivalence of the two definitions of dot product, $\mathbf{u}\cdot \mathbf{v}=u_1v_1+u_2v_2+u_3 v_3$, and the equation $\mathbf{u}\cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta$.
This is a consequence of the cosine law, if we take the triangle with sides  $\mathbf{u},\mathbf{v}$ and $\mathbf{u}-\mathbf{v}$ then the cosine law reads
$$|\mathbf{u}-\mathbf{v}|^2=|\mathbf{u}|^2+|\mathbf{v}|^2-2|\mathbf{u}||\mathbf{v}|\cos \theta$$
using
$$|\mathbf{u}-\mathbf{v}|^2==|\mathbf{u}|^2+|\mathbf{v}|^2 -2\mathbf{u}\cdot \mathbf{v}$$ we get the desired result.
