This is a pretty famous/cool game.
So...
You use the the above poster leaves behind on the island to
escape the island, and you leave something behind.

Ok, I'll start.

I leave behind a pot of steaming water.

I pour out the water and use the pot as a boat, floating to a real country. Upon doing so, I leave behind my Hawaiian shirt.

I burn the shirt as an SOS signal to attract rescuers. As I'm plucked away by helicopter I leave behind the wooden chair on which I floated to the island.

I use the chair as fuel for a signal fire and am rescued by a ship.
Now that I am about to return to society, I leave behind the volleyball I found washed up on shore and talked played with.

I rig the parts of the radio control into a transmitter and radio for help. A nearby fishing boat picks up the signal and rescues me.
I leave a bag of dog treats.

I manage to live off of dog treats while using the bag as a flag to get the attention of a helicopter.
I leave behind a postcard that says "Enjoy your stay at Hawaii"

I write a note on the postcard that tells my situation and my coordinates, then put it in an old bottle I found washed up on the beach and toss it out to sea. After a few months, a rescue party comes and finds me, none the worse for wear.
I leave behind my elaborate series of treehouses.

I take them apart and build a raft to safety, but leave behind my hammer.

I use the hammer as a weapon to hunt. Over time I build up a cache of bones and hides, which I use to make a coracle with outriggers and a hide sail. I sail to safety.

I leave a cache of preserved meat that I never got around to eating.

I use the meat to attract some sharks. I use their skin to make a raft and I use the remainder of the meat to attract some gulls that I use to tow the raft. The birds pull me so fast that my hat falls off and gets left behind.

I use the hat like I'm Oddjob to hijack a pirate ship. I then use the ship to create a massive lighthouse. This attracts the attention of a better ship with a radio, so I hijack that too. I use that radio to attract a fishing boat. I then steal a fishing rod. I catch a trout with it and then flaunt it about to the heavens. Pete decides to take the fish and eat it. Before leaving he tells me 'It's time to wake up.' I wake up back in my bed safe at home.

I leave behind a book describing my overly elaborate plan.

I use the pages to build a working computer out of paper and tell my situation in a forum post which, if you were wondering, read "Hey, doods. I'm kinda lost on this island right now. Anyone gessum halp? ". Lucky I just happened to find working Ethernet cables and a generator.

I leave behind "Footprints in the Sand" a reference to CW's Arrow. Have fun! Just 5 years to go!

I panic so badly that I swim across the ocean. I leave behind my marbles.

I trick a few fish into thinking they're food and they swim me across the ocean. I leave behind the Wind Waker baton, but it's broken in half.

I put the halves up my nose and teleport home. Because that makes perfect sense.

I leave behind Derp Ssundee. If you don't know who that is, look him up on YouTube.

After watching far too many music videos I conclude one thing. Minecraft logic takes precedence now.
I killed myself and respawn in my bed at home.

I leave behind a bucket of lava.

I use it to make a flying thing and go somewhere else with it. Unfortunately, Derp Ssundee found me and came back with me. Do you have any idea what you've unleashed?

I leave behind Justin Beiber.

Things become very un-PG-13. I'm arrested and taken back to the mainland for trial.

I leave behind a island sized game of cluedo.

It inspires a line in a Propaganda song. I'm taken back to work (at the HHMB Records studio) for production.

I leave behind all the Reach Records and HHMB Records albums that have been Grammy-nominated (which is only about a dozen, so it's not that much).

I use them like surf boards to get to the main land.

But I leave behind a giant bell.

I use the bell as a helmet and walk on the bottom of the ocean all the way back to civilization.

I leave a spare carbon dioxide scrubber.

I rig the carbon dioxide scrubber up to take the CO2 out of the air at a faster rate. Soon scientists come to investigate why the area around the island has less CO2, and they find me and take me back to civilization (Not the ferret, you know what I mean)
I leave behind a digital wristwatch that still works.

I use the watch to signal help. however I got the attention of some aliens, they offer to take me back to the mainland for the really cool watch (their watch technology isn't very advanced). They leave behind their experimental jet pack to make room for me.

I break down the only coconut palm on the island, use the jetpack for propulsion and float away leaving behind the leaves and three coconuts.

I pretend I'm King Arthur and use the coconuts to simulate horses. I'm arrested and taken back to the mainland police station. I leave behind my shield, because it's an offence weapon.

I use the shield as a surf board and surf back to the mainlands. But I leave behind...*sees shoulder is empty and feels around on head, finding nothing* MICKEY?!?!

I wait and chat with Mickey until a rescue part comes and picks us both up. I leave behind a notebook full of doodles.

'I found a Pirate's Map!' I try to follow the drawings but end up somewhere in the Sahara Desert. I left behind a ludicrous amount of footprints.

I bring home to the island and leave the world behind.

I have the whole world on the island. I roll it off the island and step off the island onto the world, landing on the island. Seeing the 'portal' effect, I jump a little further onto the world and land on the world that's next to the island that's in the world. Each time this happens I land a little further, eventually landing past the World by the island and therefore not the island.

I leave behind an imploded mind.

I pick it up, seeing if I can use it. When I see I can't, I throw it out into the sea. A whale flies up and catches it. I then use a paper pillar to launch myself onto the whale and ride it to victory.

I leave behind Atom from Real Steel. (How I got him is not important. Just bring him back to the main land in one piece cause he's not mine. )

I induce madness on the island. So when I split the Atom, the resulting explosion propels me to the mainland without harm. I then stich the atom back together.

I leave behind Sparta.

They kick me back to the mainland, where I scold you for nearly breaking a boxing robot that I don't own before returning him back to his dimension.

I have apparently left my dead phone on the island, though.

I use the phone like a pebble to skim across the ocean. It holes a frigate. The ensuring investigation brings to light the dead phone and therefore the direction it came from. Several days later I'm arrested for the 4th time and brought to the mainline for yet another trial.

I left behind my lawyer, Romo Lampkin.

My lawyer, Felix Wrong, Maybe you'll get it...maybe.saves me from the competitor.

I leave behind over-muscular DVS and North Star.

Miles comes to get them, of course. They take me back just cause.

I leave behind my copy of Splatoon.

I use it as an improvised paddle, and then sell to a museum as a sole thing that aided my escape.

I leave behind exactly 1 random IKEA furniture assembly manual.

I discover that it's a manual for building an IKEA emergency raft out of palm trees. I use it to construct the raft (complete with palm frond sails), and sail it to civilization. I did such a nice job on it that IKEA sponsors me to sail it around the world and become rich as a spokesperson for their product.

I leave behind an extra part that I didn't need.

I use the extra part to signal for help and I am rescued by a passing ship.

I leave behind a copy of Twilight.