If you're wondering why can't you get to a new island, please open a travel menu first (in the bottom left corner of the screen) and then see the messages on the cards with new islands.
"Complete quest" means that you already have the necessary level for the island but you still need to complete some quest on your Home Island. We can't say what quest it is for sure, as all players play at their own speed and the unfinished quests may therefore be different for each player. Just keep in mind that you need to complete quests on your Home Island and you will get access later. Don't worry, you won't miss the opening of the new island. Once it becomes available, the new story will begin and you will receive a new active dialogue right away.
"Level required" means that you need to reach a certain level to get access to this island. Just keep exploring your Home Island and the available islands (if any of them are open at the moment) and you will get access to new islands later.