We can understand the nature of bonding orbitals by examining how they are formed from the atomic orbitals as the atoms are brought together. Bonding orbitals are formed when atomic orbitals combine in ways that lead to predominantly constructive interference. The key feature of bonding orbitals is that the molecular orbitals have a lower energy than the corresponding atomic orbitals. Thus, the molecule (the atoms separated by at a particular small distance) has a lower energy than the separated atoms (atoms separated by a large distance).
Another characteristic of bonding orbitals is that the electron density is found between the atoms. This leads to our idea that covalent bonding is "shared" electrons. The electrons have a high probability of being between the nuclei in the molecule.
σ-bonding is due to the end-to-end overlap of orbitals having constructive interference (in phase). All σ-bonding is "on axis" meaning the electron density is centered directly between the two bonding nuclei. Below are two figures showing σ-bond formation. The first figure shows two s-orbitals overlapping to give the σ-bond. The second figure shows the end-to-end overlap of two p-orbitals to give the σ-bond.
In diatomic systems, the MO look remarkably like the bonds we think about with VB theory since the whole molecule is only two atoms (and the bond between them). Thus we use the same labels of "sigma" and "pi" bonds. In larger molecules, the orbitals will have complicated shapes but the same ideas will hold. Lower energy molecular orbitals compared to separate atomic orbitals with higher energy favors bonding.