Generators of $\pi_1(\mathbb{RP}^n)$

Question

There's a proof in Tom Dieck p.$436$ of the Borsuk-Ulam Theorem which is the following:

What I don't get here is why taking a generic path $v$ from $x$ to $-x$ we know that composed with the orbit map generates $\pi_1(\mathbb{RP}^n)$ and $u = Fv$ generates $\pi_1(\mathbb{RP}^{n-1})?$

Is there any way to see the latter algebraically or there is a simple geometric idea behind it?

Any help hint or reference would be appreciated.