Vector fields- differential topology

Question

Can anybody please explain me the reason for last 6th line. (The set of all vectors at all points...)

Answer 1

$p\in\mathbb{R}^{n+1}$ and so is $v$, hence the pair $(p,v)$ can be seen as living in $\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$ which is $\mathbb{R}^{2n+2}$.