When you see professor booster fall in the final labyrinth room, avoid speaking to him.
This is the first in a series of steps to unlock a bonus area.

To skip Professor Booster, you will need to cross the gap without falling down.
If you don't have a level 3 machine gun, you should use the little red dot on the platform as a jumping guide. It is tricky to pull off, but it can be accomplished without the gun (this is intentional)
The red guide dot. This is in the game for the sole purpose of showing you where to jump from.
If you've brought a machine gun (i don't reccomend it, it is the least useful weapon to trade your Polar Star for) then you won't need to worry about this jump.

Oh, and you can't climb back up if you fall. Pretty annoying, but in the Nintendo Switch version (other versions are available) the room has been edited and you can climb back up.

After exiting the labyrinth and escaping with Curly, please continue to enter the giant grey room. Once you make it through the robotic doors, follow Curly into the big room full of dead robots. DO NOT APPROACH HER YET, IT WILL TRIGGER THE NEXT SEGMENT. Go down to the bottom right of the room. You'll see some rubble, and a dead, rusted robot. And some shiny sparkles. If you haven't taken the booster, you will be able to loot this robot for a tow cable. Once you have the tow cable, talk to Curly. Once the Core fight is over, a cutscene will happen and the room will fill with water. You will drown (This is normal) and recieve some strange exposition. Think about what you're doing with your life. Curly will be on the floor, everybody walk the dinosaur, so interact with her by pressing the down key.

You will be given the option to attach her to the Tow Cable.
Pick "Yes".
Now she's tethered to you, and you can continue with her.

Oh, and she gave you her overpowered bubble shield that lets you breathe underwater forever.
Now go to Part 3.

The Infamous Jump

Alright, now we're cooking with gas. Remember to save. The next area is the Waterway, which plays around with the screwy water physics; good thing you've got that bubble. The main danger here is the spikes, although there is a new Jellyfish enemy here. Jellyfish bob around in the water in a fixed motion. Make sure to kill as many as you can before subjecting yourself to currents.

Once you reach the midway point of the Waterway (just past the big room of Jellyfish with annoying water currents) you'll reach a small room that contains a one-way gap to the rest of the game, and past that, a shack. This jump can be pretty annoying thanks to the water, and you don't have a Booster to save your ass if you miss. Thankfully, it's significantly easier than the one in the labyrinth. There's no shame in reloading if you miss, but you will have to do the Waterway again.

If you fall down before entering the shack and doing the event in there, Curly's life support will run out, and you cannot salvage her.

Once you get to the door, enter it and it will take you inside the Waterway cabin (great nickname, right?) Sleep on the bed (with curly) and you will regain HP. Curly will now be on the bed. Now, look at the computer. It should tell you about a journal. Look at the bookcase- it contains the journal. You will find out how to drain robots. (You cannot find the journal unless you check the computer first.) Then go and examine Curly. It will give you the option to drain the water in her. Select YES. After the water-drain process, Curly will wake up and give you some dialogue. Then she'll fall asleep. Examine her a few times, and you will be given the option to leave her.
Select NO to take her with you. You cannot return to the Waterway once you leave, so don't forget to do this. Carry her with you for the rest of Waterway. Her life support will not run out. Fight the boss*, then curly will float away from you. This is normal, continue.

*nerd bonus: this boss is a reference to one of Pixel(the developer)'s older titles.

(Go to arthur's house.)
Professor Booster will teleport in if you followed the previous steps. He will give you the Booster v2.0.
This makes life a lot easier, and you'll notice it's a lot more useful than the v0.8 you would have, had you not made the effort.

-The Booster 2.0 can be used to do the slow diagonal jump of the 0.8 by holding down and left/right while boosting. It's unwieldy at first, and may require some getting used to. I suggest testing it out and learning the feel before moving on to the next area.

Liberal use of the Booster 2.0 will be required later on, so try to get used to it and it will make things a lot easier for you.

It's not over yet!

Okay, you've saved her, but not really. Once you get to the Plantation, Jump down to the part by the lake.

Immediately down you will find a similar jump path boring down into the side of the mountain. If you saved Curly and successfully drained her of water, you will find her bedridden at the bottom room, next to a Cthulu (the weird green guys) who found her. Chat with the Cthulu, then Curly, and then the Cthulu again (yes you have to) to learn that Curly has amnesia now. Great.
You also learn that there's a mushroom which can restore memories somewhere on the island.
It's all the way back in the Mimiga Village Graveyard.

Don't groan yet! There are two teleporters in the plantation (as well as other goodies, make sure to have a look around) that will make this a lot easier. I mean, you don't want to go back through the Outer Wall and Egg Corridor(?) again, right?

In the upper left area of the Plantation, you will find Jail Cell No. 2. The teleporter in here (break the wall on the left to reach it) will take you to Kazuma's shelter in the Bushlands. Traverse the Bushlands and return to Arthur's House via the teleporter near Santa's House.

There is another teleporter that goes straight to Arthur's house, but it requires a key to access (and some story). If you go to the Rest Area (guarded by one of the little green bud enemies, fairly harmless) you'll find the fishing mimiga from Mimiga Village. Talk to him, and he leaves. Meet him at the lower portion of the Plantation (the big lake) to recieve a key to the teleporter room. The first time you attempt to use this teleporter, you will be caught and thrown in Jail Cell No. 1, where you will recieve some story and exposition. After you leave the cell, you will be able to return to the Teleporter Room, and gain an instant travel point between Arthur's House and the Plantation.

In Mimiga Village Cemetery, there is a door at the top right that is inaccessible without a booster or level 3 machine gun, but is only useful if you're already trying to save Curly. Speak to the mysterious blue mushroom. He'll offer you a badge.

Say yes when he asks you if you want it, and no when he asks if you don't want it. If you mess up, you'll have to try again. Eventually, he'll give you the damn badge. Don't leave the room.

Open your inventory, and check the badge. It's absolutely useless. Talk to the mushroom, and you get the easiest boss fight in the game*. After you kick its ass so hard it doesn't move anymore, you will finally have the damn mushroom you need to fix Curly's memory.

*unless you suck

Go back to the Plantation, go all the way down to Curly's bed, and feed her the mushroom.
After a lengthy chat that reveals a lot of important story stuff, she'll give you the Iron Bond.

Once you have the Iron Bond, you've got access to the bonus level.

In the sequence after the Balcony boss fights, the prefab shack door will be open. Normally it is closed during this part. Enter the shack (be careful not to get smashed), heal, and save. (note: in the Switch version, you cannot heal here.)

To begin the bonus level, simply jump down the hole.
If you ever wish to simply obtain the normal ending of the game, check the bookshelf. It will turn back time to before you entered the shack, but you do not lose your save in the shack.

You're welcome!