I'm having trouble understanding why anti-orbitals form.

For example:

s (H) + s (H) -> σ (H2)

 

Then the anti-orbitals in He2 (Theoretically)

s^2 (He) + s^2 (He) -> σ + σ* (He2)

Remember orbitals are just probability distributions of electrons. Anti-bonding orbitals are just out of phase linear combinations of atomic orbitals. While bonding orbitals are in phase combinations. This is more obvious with p-orbitals beacause they have more obvious lobes and nodes.

out of phase

 

I've heard that said a lot about molecular orbitals. What does it mean to be "out of phase"?

Probabilities in MO theory come from the Schrodinger equation. The wave function is the object that represents electrons in bonds according to the Schrodinger equation. The wave function is 'complex-valued', which means that at any point the wave function has both a magnitude and a direction, like an arrow pointing from the center of a circle. When two wave functions are added together, these directions come into play. At one point both wave functions might be pointing in the same direction, or 'in phase'. Then they would add directly to one another. At another point they might be pointing in opposite direction, 'out of phase'. They would be subtracted from one another. In general, wave functions add together somewhere between these two extremes.