What determines which opponents appear on my islands?

Matchmaking is level-based. Players are matched by comparing their individual Ascension Levels.

In the beginning of the game, there will mostly be easier opponents whose level is the same as yours or even a bit lower. As you level up and unlock more island areas, the game will start providing you with opponents whose Ascension Level is higher than yours, which means that their defense is potentially stronger, making them harder to beat.