A nice broad question to set the scene
I'm fairly new to roleplaying and I'm struggling with a valid reasons for both why and how a character would multiclass into another class independently – that is, on their own, without specific training from an NPC, temple, college, etc.
More detail
I'm currently playing a Rogue and I would like to multiclass into Monk, taking more levels in both Rogue and Monk in the future. The current campaign has not included a local monk temple/monastery where my character could seek training (to explain his new skills) and at this point in the campaign my character simply doesn't have time to sit in a cave for a few months learning from a (miraculously) passing grandmaster.
I'm familiar with the idea of classes as a metagame concept, so I can get behind the idea of new class abilities retroactively being part of my character's existing training. However, the sudden jump in abilities from monk levels – especially the AC improvement from Unarmored Defence – seem like such a huge change for a rogue.
Most other classes 1st level abilities make sense to an extent, and can be explained away as a gradual improvement of skills that eventually "click" and makes a noticeable difference to the character. However, I can't quite resolve why someone used to running around in leather armour would decide they'd do better without it? Or why a rogue used to stabbing people in the back would start (effectively) lashing out with fists, elbows and kicks.
Paraphrasing the PHB slightly, Monks are portrayed as ascetic communal hermits, training for years to achieve personal perfection through the study of Ki. Multiclassing into the Monk class seems to ignore this whole background concept almost entirely.
So what roleplaying options/ideas are available to me in this scenario to explain my character's sudden acquisition of new skills?
Updated Update:
For more character background detail (as discussed in the comments below) my character concept so far is a wood elf Assassin Rogue/Shadow Monk. I'm not aiming for a stereotypical street-wise city thief – instead I'd them to be more of a sneaky forest dweller (due to being a wood elf), killing only when necessary and intent on personal training and perfection. A sort of good/neutral "nature's ninja" if you will.
My character may also be taking a single level in Fighter at some point but this is a lot easier to explain simply through a personal focus/dedication to general martial training.
Extra credit
Bonus points are available for relevant or interesting concepts and ideas based around other classes outside this scenario, however, my main focus is explaining the Rogue -> Monk transition.
Best Answer
Your PC can invent a new technique on his own
Taken from wikipedia's entry on the Southern Praying Mantis kung fu style:
If your rogue is observant enough - he may become inspired by some natural phenomenon, gaining insights about efficient and effective motions useful for offense, defense and maneuvering.
He doesn't need to be an acetic philosopher (though he may grow to become one as he advances) - he just needs the conviction and self-discipline to push him to go through the physical training required to hone his technique.
If your group's play style allows placing more emphasis on narrative at the cost of poorer combat optimization, you can even select to make the transition of your PC's reliance on his new martial skills gradual - continue wearing armor, combine armed attacks with unarmed strikes (not necessarily in the same round, though), etc. But, eventually your PC will start feeling that his armor is limiting him, preventing him from escaping blows that using his technique he can now avoid - then he may consider giving up on wearing armor. Same goes for unarmed strikes - have him hold on to his dagger, but occasionally use a flurry of blows(1) using a fist and an elbow of his other, empty hand - and let him realize that his hands have become faster and more accurate and than any weapon he is used to. This can lead him to relinquish the use of melee weapons, to use an interchanging combination of weapon and unarmed strikes or perhaps even seek "better" weapons which are more compatible with his developing abilities.
Bottom line - just because the "metagame concept" says your PC can now do something, doesn't automatically means he is aware of that and trusts himself enough to give up his old ways completely and immediately.
Or, he can find written records of a technique
Taken from wikipedia's entry on the Northern Praying Mantis kung fu style:
Aside from self mastering a technique, your rogue may find a documentation of a certain technique - this can come instead of having him invent it from scratch, or as a latter supplement giving him more inspiration and advanced abilities.
At either case, you can probably work the narrative details with your GM so they won't clash terribly with the campaign settings (assuming that having a monk in it doesn't do that in the first place).
(1) Or whatever equivalent term used in 5e for that...