Let $T:M^m \rightarrow N^n$ be a submersion where $n \le m$.
It is well-known, in this case, that a foliation (called a simple foliation) of dimension $m-n$ (or codimension $n$) is defined.
Its leaves are the connected components of the fibers of $T$ (i.e., the connected components of $T^{-1}(x), x\in N^n$).
The question is: Is not it assumed that $T$ surjective? (to make sure that $T^{-1}(x)$ is not empty for all $x\in N^n$).
You either assume it is a surjective submersion or, what is usually done, is that you can take for granted that $x$ lives in the arrival space $T(M^m)$ of $T$. So formally you have $T:M^m\rightarrow T(M^m)\subset N^n$.
This is perfectly coherent with the definition of a foliation of $M^n$, as a collection of disjoint, connected, non-empty, immersed $(n-m)-$dimensional submanifolds of $M^n$. Indeed, it is enough to restrict $T$ to its range in order to define such a foliation.