Whenever a spell or ability resolves, the active player receives priority. This matters greatly with regard to planeswalkers. This is because immediately after a planeswalker spell resolves, its controller can choose to use one of its abilities before any other player will get a chance to respond. If the cost to use the ability increases the number of loyalty counters on the planeswalker, then by the time another player receives priority, this planeswalker will have additional counters on it.

For example, immediately after Liliana of the Veil resolves, she will have three loyalty counters on her. Her controller can then use Liliana’s first loyalty ability at that time. Doing so will take her up to four loyalty counters because the cost of adding a counter is paid as the ability is put on the stack. Even if an opponent had Incinerate in hand, it would not be able to deal enough damage to Liliana to cause her to go to the graveyard. This is because by the time this player can play the Incinerate, Liliana would have already four counters on her. Even if Liliana were killed with one of her abilities on the stack, that ability would still resolve normally because it exists independent of its source like all abilities do.