$c \equiv b \pmod{a}$ is a short hand notation to denote $a \vert (c-b)$ (or) equivalently $b$ is the remainder when $c$ is divided by $a$. Typically, for convenience people have $b \in \{0,1,2,\ldots,a-1\}$.
In your case, you want to evaluate $-5 \pmod{4}$. By that I assume you want to mind $b \in \{0,1,2,3\}$ such that $-5 \equiv b \pmod{4}$ i.e. we want to find $b \in \{0,1,2,3\}$ such that $4 \vert (-5-b)$. Since $4 \vert (-5-b)$, we have that $4 \vert (5 + b)$ and hence $4 \vert (1+b)$. Hence, we get that $b=3$.
Therefore, $-5 \equiv 3 \pmod{4}$
To interpret $b \pmod{a}$ as a distance, in the sense you mean, it is the distance of $b$ from the largest multiple of $a$ no larger than $b$ i.e. the largest multiple of $a$ that falls to the left of $b$ or $b$ itself on the real number line. I have tried to illustrate this with couple of diagrams below.
The first picture indicates $b\pmod{a}$ when $b$ falls between $4a$ and $5a$. 
The second picture indicates $b\pmod{a}$ when $b$ falls between $-3a$ and $-2a$. 