The diagram you shared is a molecular orbital diagram. You might have learned in your class about molecular orbitals (MOs). Briefly: just like how atoms contain their electrons in atomic orbitals, MO theory says that molecules contain electrons in molecular orbitals. These molecular orbitals are built by various combinations of atomic orbitals, which are contributed by the atoms that make up the molecule. The approach of building molecular orbitals from atomic ones is called the linear combination of atomic orbitals (LCAO) method.
In the theoretical molecule di-helium, each helium atom contributes its 1s orbital to the molecular orbital configurations. MO theory says that the number of MOs made is equal to the number of atomic orbitals (AOs) put in. For di-helium, two AOs are put in (the 1s orbital of each helium atom), so two MOs are created. Of these, one MO will be lower in energy than the AOs are, and we call it the bonding MO. The other MO will be higher in energy than the AOs, and we call it the anti-bonding MO.
Here's an important distinction that your diagram is inaccurate on: due to the quirks of quantum mechanics, the energy difference between the anti-bonding MO and the AOs is larger than the energy difference between the bonding MO and the AOs. In other words, the anti-bonding MO is more destabilized relative to the AOs than the bonding MO is stabilized relative to the AOs. Your diagram inaccurately describes both energy differences as 104 kcal.
The fact that the anti-bonding MO energy difference is larger than the bonding MO energy difference is the true reason that di-helium doesn't exist. Of the four valence electrons in di-helium, two will fill the bonding MO, and the other two will fill the anti-bonding MO. The two electrons in the bonding MO will achieve some stabilization relative to their position in the AOs, but the two electrons in the anti-bonding MO will achieve greater de-stabilization relative to their position in the AOs. The net result is a less stable molecule than if the electrons remained in their respective AOs.
Compare this to di-hydrogen. The same AOs are contributed as in this di-helium example, and the same MOs are made, but in the case of di-hydrogen there are only two valence electrons. The two electrons fill the bonding MO, while the anti-bonding MO remains empty. The net result for di-hydrogen is a stabilization of the electrons relative to their position in the AOs, and di-hydrogen is preferable to isolated hydrogen atoms.