I had the following idea: to prove this by induction.
Assume that we can prove it for a one-manifold. Now take a two-manifold $M$, and embed it in $R^5$. Take two points $p,q\in M$. Then there exists a hyperplane $H$ in $R^5$ containing $p,q$ such that $H\cap M$ is a connected one-manifold. Hence, by the inductive hypothesis, we are done. For all manifold of higher dimensions $n$, we just have to ensure that there exists a hypersurface that intersects the embedded manifold such that the intersection is a connected $n-1$ manifold.
Is there a way of making this proof work?
Perhaps that it can work, but that approach looks too much complicated. Besides, there are $2$-dimensional manifolds which cannot be embedded in $\mathbb{R}^3$ (the Klein bottle is an example).
The statement is easy to prove using the fact that manifolds are locally homeomorphic to $\mathbb{R}^n$, together with the fact that, in $\mathbb{R}^n$, open connected sets are always path-connected.