Formation of Bonding and Anti-bonding Molecular Orbitals
First, remember that orbitals are a more or less theoretical construct, based on mathematical models. With that in mind, remember that an orbital in the strictest quantum-mechanical sense is a wave function of one and exactly one electron. (I am separating &#x3b1; and &#x3b2; spins here.) Since an orbital somehow nicely corresponds to one possible particle state, it may seem logical to postulate a ‘rule of conservation of orbitals’ — and exactly that happens in quantum chemistry.

So if we take one 1s orbital of one hydrogen atom and another 1s orbital of another hydrogen atom, we need to create two new orbitals by linear combination if we choose to do so. The two orbitals we create correspond to &#x3c3; and &#x3c3;*, respectively. (In a spintastic sense, we would have taken two orbitals of each hydrogen, one corresponding to &#x3b1; the other to &#x3b2;. We would receive four orbitals: $\unicode[Times]{x3c3}_\unicode[Times]{x3b1}, \unicode[Times]{x3c3}_\unicode[Times]{x3b2}, \unicode[Times]{x3c3}^*_\unicode[Times]{x3b1}$ and $\unicode[Times]{x3c3}^*_\unicode[Times]{x3b2}$. The electrons now occupy the lowest energy orbitals which are $\unicode[Times]{x3c3}_\unicode[Times]{x3b1}$ and $\unicode[Times]{x3c3}_\unicode[Times]{x3b2}$.)