Right now im working on a proof in the Paper Eight faces of the Poincare homology 3-sphere (page 124-28,13-16) and the main aspect of the proof is how the multiplication of $\mathbb{S}^3$ geometrically works. Most of the explanations I can understand, but I really don't know how one has to understand the red marked excerpt in the picture below (especially what perpendicular in this context means.
I'm not sure what you want to know about the perpendicular sphere, but since the "extrinsic" description using perpendicularity is unsatisfying to you, here is an intrinsic description.
First, the given point $p$ has a unique antipode, denoted $-p$, characterized as the unique point in $S^3$ of maximal distance from $p$, that maximal value being $\pi$.
Second, the perpendicular sphere of $p$ is the set of points located exactly half-way between $p$ and $-p$, that halfway distance being $\pi/2$.
There is an analogy one dimension down, in the 2-dimensional sphere. Again each point has a unique antipode, and it has a perpendicular circle which is the set of points halfway between the given point and its antipode. In this analogy, think of $p$ as the north pole, $-p$ as the south pole, and the perpendicular circle as the equator.
Thus, going back up to the 3-dimensional sphere, one can think of $p$ as a pole, $-p$ as its opposite pole, and the perpendicular sphere as the equator with respect to those two poles.