There are strictly increasing $C^1$ functions that map sets of positive measure to sets of measure zero. Here is a construction:

Let $C\subset [0,1]$ be a Cantor set of positive measure. For a construction, see https://en.wikipedia.org/wiki/Smith-Volterra-Cantor_set. Let $g(x)=\operatorname{dist}(x,C)$. The function $g$ is clearly continuous and equal zero on $C$. In fact $g$ is a $1$-Lipschitz function. Let
$$
f(x)=\int_0^x g(t)\, dt.
$$ 
The function $f$ is $C^1$ and it is strictly increasing. Indeed, if $y>x$, then
$$
f(y)-f(x)=\int_x^y g(t)\, dx>0
$$
because the interval $[x,y]$ is not contained in the Cantor set $C$ and therefore it contains an interval where $g$ is positive.

On the other hand $f'=g=0$ on $C$ which has positive measure and $f(C)$ has measure zero since $m(f(C))=\int_C f'(t)\, dt=\int_C g(t)\, dt=0$.