I am looking for the original proof by Borsuk of the Borsuk-Ulam theorem. I would appreciate very much if someone could outline the proof.
As J.C. Ottem suggests, the best thing in those cases is to look for the original paper. Wikipedia says
According to (Matoušek 2003, p. 25), the first historical mention of the statement of this theorem appears in (Lyusternik 1930). The first proof was given by (Borsuk 1933), where the formulation of the problem was attributed to Ulam.
Here (Borsuk 1933) is the paper
Drei Sätze über die n-dimensionale euklidische Sphäre, Fund. Math. 20: 177–190,
and a digitalized version of it can be found at this link.