In Guillemin & Pollack Differential Topology it says on P.49 with regards to Whitney's weak embedding theorem (i.e. every smooth $k$-manifold can be embedded in $2k + 1$ euclidean space):
Why $N = 2k + 1$? As we see it, the underlying geometrical reason why $2k + 1$ space can accomodate all $k$-dimensional manifolds is a basic property of transversality that will be proved in the next chapter.
Unfortunately, they never really talk about the topic again. At least not explicitly. From what I've read I suspect that some dimension counting argument is involved, but I really don't know which of the theorems in their chapter on transversality ensures Whitney's weak theorem the needed wiggle room in $\mathbb{R}^{2k +1}$. I would really appreciate it if someone could maybe explain their thoughts a bit more.