The contrapositive of $A \rightarrow B$ is $\lnot B \rightarrow \lnot A$.
In your example, we have to "parse" the complex sentence according to the "if _, then ..." form.
We have that :
$A :=$ "$f : X \rightarrow Y$ is a continuous surjection bewteen separable compact metric spaces, where $X$ has dimension $n$ and $Y$ has dimension $m$"
and ;
$B :=$ "there is a point $y \in Y$ whose preimage contains at least $n−m+1$ points".
Thus, the contrapositve will be :
if not "there is a point $y \in Y$ whose preimage contains at least $n−m+1$ points, then not "$f : X \rightarrow Y$ is a continuous surjection bewteen separable compact metric spaces, where $X$ has dimension $n$ and $Y$ has dimension $m$"
or, in a slightly more palatable form :
if "not exists a point $y \in Y$ whose preimage contains at least $n−m+1$ points, then "$f : X \rightarrow Y$ is not a continuous surjection bewteen separable compact metric spaces, where $X$ has dimension $n$ and $Y$ has dimension $m$".
