There is, probably, no hope for a similar characterization since for any topological space $Y$ and any $X$, which is line-filling in your terminology, if $f: X \to [0,1]$ is continuous and surjective then $h: X \sqcup Y \to [0,1]$, where $X \sqcup Y$ is a sum of spaces (=disjoint union) $X$ and $Y$, given by $h|_X = f$ and $h(Y) = 0$ is also surjective and continuous. So your spaces maybe as bizarre as you want.
Update: Alejandro, thank you for your comment. I don't think though, that connectedness (or path-connectedness) would make things better. Again, for any connected (path-connected) Y and any line-filling space $X$, $f: X \to [0,1]$ take $x \in X$ and $y \in Y$ and glue $X$ and $Y$ at the point $(x,y)$ (that is consider equivalence relation with only one non-trivial equivalence $x \equiv y$ and take the factor space). Denote the result by $Z$. Then one extends $f$ to $Z$ by defining $f(Y) = f(x)$. I believe $Z$ is then connected (if $X$ and $Y$ were) and again pretty random.