To steal a phase means that during that player’s next turn, as the chosen phase begins, you treat it as if it were your turn. Here’s what happens if you steal each phase:

Beginning phase: You untap permanents you control and your opponent doesn’t. Abilities that trigger at the beginning of your upkeep happen and ones that trigger at the beginning of their upkeep don’t. You draw a card as it’s now your draw step, so your opponent doesn’t.

Precombat main phase: You can cast all types of spells while your opponent can only cast instants and spells with flash.

Combat phase: Just like combat on your turn, you can attack your opponents with your creatures. The opponent you stole the phase from can’t attack with their creatures, but they can block.

Postcombat main phase: Same as precombat main phase.

Ending phase: Most abilities that trigger during the end step don’t care whose turn it is, but a few do. Effects that last “until end of turn” or “this turn” will expire as normal, as they also don’t care whose turn it is.