After a while in Disco Elysium it will become clear that there's only one more course left to you - make your way to the final possible location of the shooter. We'll show you how to get there, and to the island in the bay.
Warning: the following contains spoilers for the plot of Disco Elysium (though we won't tell you what's ON the island, of course).
What Day You Can Go to the Island
Earlier in the case you'll establish that the murderer could've been working from any one of four locations, but exactly which one has been a mystery. The island has been a low chance so far, but now it's the only option, made more plausible by the evidence Klaasje left in her room. This means you can't actually get there until after the tribunal, which is likely to be around Day 7 (though individual playthroughs may change this).
How to Get to the Island
Once you check Klaasje's room and get the task to go to the island, head across the water lock to the Fishing Village and talk to Liliene the Net Picker, the dark-haired woman in yellow standing near the pier. Talk to her about her boat, and explain you need to get to the island. She'll let you use the motorboat tied to the end of the pier, now all you've got to do is click on it.
Warning: if there are things you want to do in Martinaise, Land's End, or the main land in general - do it now. You won't get another chance after you climb into the boat.
If you found that helpful, you can check out our guide to finding all the game's ceramic armor at our guide here, or check out our guide to the game's length over at this link here.