Did he just arrive on Mêlée Island™? If so, where did he make landfall? At Stan’s? But how did he get past the bridge troll then?
Or was he living on the island, has even been born there? Where is Guybrush’s house then? Lost on disk 22?
Seems like there is more than one Secret™ in this game!
There would be lots of to speculate. 
If he just arrived, he is coming from Stan’s, but he arrived at night, probably sleeping on the ship for a days. How old is he? 17? He left his family months ago to follow his dreams. His home island is not on the map.
He past the bridge before George Lucas stepped on it.
The island has many other shores he could cast away and woke up days later.
Before he was Guybrush and arrived on the island he used to be Jeff Bridges but then got scanned to the game grid.
Well according to the intro cinematic of “Escape from Monkey Island” (if you count that game as canon) he states that “it’s hard to believe it’s been a few years since I washed up on the shores of Melee Island.”
So he just appeared out of thin air…
Or you could speculate that his ship sank, and he decided he wanted to be a pirate at that moment.
O come on guys!
It’s so clear! He got there as an illegal immigrant. After wandering homeless and jobless, he moped so much he convinced himself the only occupation he could get was to join organized crime. 
Well, do we want to go with the “it was all a child’s play” interpretation?
Guybrush just “appears” there. It’s the moment where they start playing. He’s Guybrush, and he wants to be a pirate. It just starts there. Before, he was in the “real world”, being a kid and all. Oh, if only we got to see the third game.
If we don’t, well, he… uh… parachuted.
But the real question is: how did Guybrush get LeChuck’s beard if we see his ghost head launched to the sky with the beard still attached to the face?
Are you implying he might have arrived with
The Caravan?
“Beam me down Scotty”
“Aye aye, Guybrush”
I would, at least. While I took MI1 (and even 2) quite literal when I played them as a kid, I now believe that Guybrush is a kid at a (possibly pirate-themed) theme park, indulging in his fantasies. LeChuck must be his older brother (we all know, from hearsay or experience, that older siblings are always mean towards the young ones, right?).
Anyway, to answer the question, Guybrush just entered the park through the main gate (represented by the stone arch).
But who knows, maybe he’s just a poor guy tied to a chair somewhere in a lab, subjected to cruel brain experiments that cause funny “dreams”.
Does that make Monkey Island a Battle Royale game? 
Well obviously Guybrush picked it up later and put it on a lance outside his yard to keep Yehova’s Witnesses away…as one does, duh!
Then he put the beard in his pants…for comfort… 
I like that symbolic interpretation A LOT actually!
And I´m not easy to impress…oh wow look, a blue car! 
Well, a dead ghost pirate’s beard sure makes for good underwear.
Very very nice reference, my yellowloving friend.
That´s the cue for @gffp to put that sentence on a Calvin Klein ad or something…
Excellent!
He was dropped off near the mountain top with a chopper. Obviously!
What goes up, must come down.
Who put that up there, by the way?
Al the (Stone) Archer?
Stop it, you reminded me of Call of Duty: Black Ops. 
(Isn’t there a yawning smiley?)
In Battle Royale you are gassed and wake up on the battlefieldMêlée Island.
Well we’re shown in EMI that he washed ashore Melee Island before walking up to the lookout in SMI, and he doesn’t seem to be familiar with the Tri-Island Area or Gulf of Melange, so he’s probably from another island group or from the American colonies or something. I like to think he was working for a company as a fisherman but was never content with the job (as in the never-made, official, non-canonical Monkey Island movie), and perhaps a storm sunk his fishing ship one day, leading to him end up on Melee. Perhaps he had always been interested in pirates all his life, and so his washing ashore in a new land was the perfect start for a new life for him to become one, or maybe he was inspired by something he saw or something that happened before his ship sunk.
I had always assumed that he set out for Melee, knowing full well that it was a pirate nest, because announcing to random islanders that you want to be a pirate doesn’t sound like such a clever idea. OTOH, washing up on a pirate-infested island first and then thinking about a career change isn’t so stupid either 
.
Though I still have the impression that it must be the former, because he’s really pouring his heart into the whole ordeal.
I got a feeling he thought a pirate infested island called Melee would be the best place to learn some melee ( =close contact = fencing) combat techniques. 
Or, he is just an illiterate boy, not uncommon in those days and was looking for the second-greatest-Legend of all time after Big Whoop when he washed up on Mêlée’s shores: the Real Mêllée Vanêllée!
Before washing up on the shores of Mêlée Island [1], declaring that he would like to be a pirate [2], and making the hike to the Lookout Point seeking instruction in his chosen craft of pirating [3, 4], Guybrush Threepwood was from the Old World [5]. He had been abandoned by his parents (or so he thought; it may have been their deaths that separated them from him) [6, 7], but not before Guybrush’s mother had had time to try to get him to practice the piano, which he would later regret that he did not ever do [8]. But he was likely in an orphanage from a very young age [9].
It would become naive, young Guybrush’s life’s ambition to become a fierce, swashbuckling, bloodthirsty pirate [10]. He may have even lived on his own in a miniature-pirate-ship-styled houseboat and dressed in pirate costumes, discontent with his job, which he would have daily threatened to quit (only to not be taken seriously by his boss, of course) [11]. And during the Golden Age of Piracy in the late 1600’s, pirates were secure in their Caribbean strongholds, amassing huge fortunes from plundering shipments of New World gold; these tales of pirate wealth thus attracted Guybrush, as well as many other adventurers [5], and his travels led him – fresh from the Old World – to Mêlée Island (somewhere in the Caribbean) in a quest to fulfill his dream [5, 10].
But something must have caused the ship Guybrush was on to sink before he was able to safely reach Mêlée’s harbor in it, resulting in his washing ashore, leaving him alone in a new land where no one knew who he was or where he had come from, and with no money [1, 3, 5]. But what he did have were high hopes and an insatiable curiosity and desire to become a pirate [5, 12], though he was not yet truly aware that there was more to being a pirate than simply swinging a sword and drinking grog [10]