Answer
The Pauli exclusion principle demands it.
Work Step by Step
No two electrons are allowed to occupy the same quantum mechanical state in the helium atom. In the ground state, both electrons have n = 1. Then there is only one possible value for the orbital quantum number, l = 0, and only one possibility for the orbital magnetic quantum number, $m_l=0$.
The two electrons cannot share every quantum number, so the fourth possibility, spin, must differ. One must be "spin up" and the other must be "spin down".
