What is responsible for making the bonds identical?
The hydrogens are all equivalent. When thinking about MOT, you have to throw almost everything you thought you knew about bonding out the window. The electrons are not localized between any two pair of atoms—they are truly molecular orbitals. This graphic nicely shows the phasing of the molecular orbitals of methane:

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The lowest energy orbital is a linear addition of the carbon $\mathrm{2s}$ orbital and all four of the hydrogen $\mathrm{1s}$ orbitals. The pair of electrons occupying this orbital is not specific to any one hydrogen, and bonds all the atoms in methane together (even the hydrogens to the other hydrogens). The other bonding orbitals are of degenerate energy and result from linear combinations of the carbon $\mathrm{2p}$ orbitals and the hydrogen $\mathrm{1s}$ orbitals. 

Naturally a molecular orbital that results from a combination of a $\mathrm{s}$ orbital will be of lower energy than one than one that results from a $\mathrm{p}$ orbital.