The enchanted player's additional beginning phase will be a lot like their normal beginning phase. During the untap step, permanents will phase in or out as appropriate and they'll untap their tapped permanents. Anything they control that triggers "at the beginning of your upkeep" or similar will trigger during their upkeep step, and they'll eventually draw a card for their draw step.

The additional beginning phase all happens during the current turn. Any effects that last "until your next turn" or similar won't expire just because they'll go through an additional beginning phase.

After the additional beginning phase, the game proceeds to the ending phase (unless something has added even more phases; see below).

If multiple phases are added to the same point in your turn, the most recently created phase happens first. For example, say you are enchanted by Shadow of the Second Sun and its ability triggers during your postcombat main phase. Later during that same main phase, another effect gives you an additional combat phase after this main phase. The additional combat will happen first, followed by the additional beginning phase.

Even if the enchanted player doesn't attack with any creatures during combat, they'll still get a postcombat main phase and Shadow of the Second Sun's ability will still trigger.

If the enchanted player somehow has more than two main phases in a turn, each main phase other than the first one is a postcombat main phase, and Shadow of the Second Sun's ability triggers at the beginning of each of them.