What does it mean to have an absolute value equal an absolute value? I have no problem reading absolute value equations such as $|x -2| = 2$. 
I know this means that the distance of some real number is $2$ away from the origin. Because the origin splits the number line into a negative side and positive side then the numbers inside the absolute value symbol will be $2$ and $-2$, since those are the only two numbers $2$ units away from the origin. Then, it's just a matter of finding the values of $x$ which will give $2$ and $-2$ inside the absolute value.
Therefore, $|x - 2| = 2$ which is 
$x - 2 = 2$ 
or 
$x - 2 = -2$
And the solutions are $\{0, 4\}$
But when I see $|3x - 1| = |x + 5|$, I have no idea know what this means. I know how to solve it, but I don't know how this relates to the distance from the origin or how to interpret this on a number line. My initial interpretation is to say, "the absolute value of some unknown number is the absolute value of some unknown number," but that doesn't tell me the distance from $0$. 
My Algebra textbook gave the following definition: 
If $|u| = |v|$, then $u = v$ or $u = -v$. 
But I can't really tell why this is the case. 

Your interpretation is good. 
Any value $v$ is at distance $|v|$ from origin. Sometimes we are given the distance and are asked to find original value. When something ($\in \mathbb{R}$) is at distance $|w|$ from origin, it has a value either $w$ or $-w$.
As you said, $|x-2|=2$ means that $(x-2)$ is at distance $2$ from origin.
The same goes for the example that confuses you;
$$|3x-1|=|x+5|$$
means that $(3x-1)$ is at distance $|x+5|$ from origin.
And what can we conclude from this?
That the value of $(3x-1)$ is either $(x+5)$ or $-(x+5)$, and that is what your textbook says using $u$ and $v$.
[Also, you can flip it and say that $(x+5)$ is at distance $|3x-1|$ from origin, and those are the redundant cases you see mentioned in other answers.]

And to directly refer to your title question; Having two absolute values equal means, in terms of distance from origin, that they are both equaly far from origin.
So $$|u|=|v|$$ means that $u$ and $v$ are equaly far from origin. How far specificaly? Exactly $|u|$ (or $|v|$, because they are equal).
