A starting point is because when you take linear combinations of two eigenfunctions of the Hamiltonian, both the (normalized) additive and subtractive combinations are also solutions to the Schrodinger equation.
A less rigorous but also less mathematical explanation is that the sign/phase of each wavefunction is arbitrary, so when you add two, they can either add positively or negatively. This is very easy to see in the case of p-orbitals that are aligned along the same axis. If we designate the phase alignment of a single p-orbital as {+·-}, where the · is the node at the nucleus, then it is arbitrarily either {+·-} or {-·+}. This makes no difference - the orbital has the same energy in free space. When two are brought together, though, they can either be {+·-}{+·-} or {+·-}{-·+} or {-·+}{-·+} or {-·+}{+·-}. Scenarios 1 and 3 are equivalent and result in destructive interference and 2 and 4 are equivalent and result in constructive interference. This is the basis for the energy differences, because 1/3 and 2/4 are no longer equivalent states of mixing. In the latter case, the wavefunctions destructively interfere with each other between the nuclei, interpreted as the electrons being located most of the time at the opposite sides of the putative molecule. This leaves the two positively charged nuclei with no negative counterbalance between them, so they repel. In energetic terms, there is an unfavorable Coulombic interaction between the two positive nuclei. This is the repulsive "antibonding" state. In the 1/3 scenario, the wavefunctions constructively interact between the nuclei, so most electron density is in this space. The positive Coulombic interactions between the two nuclei and the two electrons is more favorable in this case, which leads to the energetically stabilized "bonding state".
There's no discussion of electron exchange in this simplified discussion - and that's a big part of the energy stabilization/destabilization - but at least it should help you see why the combination of two orbitals leads to both a stabilized and destabilized (relative to isolated orbitals) delocalized molecular orbitals.
(Mixing of multiple orbitals or orbitals with different angular momenta is more complex but essentially the same principles are at work.)