FIRST PROTONATION

The lone pair on the carbanion formed from deprotonation can resonate with the nearby carbonyl group, resulting in a planar structure at the former stereocenter. Thus, the acidic proton on methanol will have access to either side of the alpha carbon near the carbonyl group (which had the carbon non-bonding electrons, the HOMO of the molecule), since the HOMO  on the carbon is symmetric from both sides of the molecule. The side that methanol protonates will form the most stable product, in which the large substituent on the cyclobutane ring is trans to the other substituents and steric interactions are minimized. Since the product of epimerization is a more stable product, the conversion of the stereocenter so that the substituent is trans to the other substituents is thermodynamically favored.

SECOND DEPROTONATION

In the second deprotonation, the carbon alpha to the carbonyl group on the ester subsitutent has an acidic proton which will be transfered to the highly basic methoxy anion. Since the resulting carbanion is stabilized by resonance (resulting in a lower energy  molecular orbital), the proton on the carbon was acidic enough to be deprotonated. Due to the high basicity of the methoxy anion and the favorable acidity of the alpha carbon proton, deprotonation will readily occur.

SECOND PROTONATION

The electrons on the carbanion formed from the deprotonation reaction can resonate with the nearby carbonyl group, and a planar structure for the carbanion will maximize orbital overlap and resonance. The relatively acidic proton on methanol can protonate the carbanion (which contains the HOMO represented by the carbon non-bonding electrons) on either side of the molecule. Methanol will protonate the molecule such that the substituent attached to the protonated carbon will be cis to the two neighboring substituents on the cyclobutane ring. The structure will be more stable than the all-cis product, since the repulsions from steric interactions will be minimzed. Thus, the thermodynamically favored product is the final product of epimerization, in which each substituent is trans to its 2 neighboring substituents.