My grandfather was a diehard Lithuanian patriot who used to tell me stories of the fatherland when drunk.
I don't have a lot of specific books on hand and these claims cover a few hundred years, but I can invoke neighbourly history textbooks, of course!
Let's dive right in!
The claim of "saving the world" is quite hyperbolic at best or false at worst. Lithuania most certainly was in the way of the Golden Horde, but it's hard to say exactly if it was a matter of overextension or actual fierce resistance. Lithuania actually got subjugated for a time.
Of course the Golden Horde wasn't the most stable of states, so Lithuania quickly pressed its advantage when it finally started falling apart to grab a large chunk of modern Ukraine. An important thing to note is that the Golden Horde wasn't the same as the Mongol Empire which was quite a different beast!
So while Lithuania certainly impeded Mongol progress, perhaps even quite effectively, saying that it "saved the world" is quite exaggerated. After all, there were invasions from the east before and the world never ended. As for the invaders, they usually ended up settling somewhere in Europe and becoming European.
Now, as for the "single-handedly" claim, well, the Polish, Belarussians, Russians, and the Ukrainians might take issue with that. Indeed, most of Lithuania was actually culturally Ruthenian. Admittedly, the Grand Duchy as a state managed to hold them off for a long time, but it eventually found allies in Poland.
And while Lithuania certainly held out against the Golden Horde, the final death blow was dealt by Muscovy once they got their act together and unified the Russian (rather than Ruthenian) principalities. Remember how I mentioned the Golden Horde falling apart from within? Well, while Lithuania played a part in it, later with Poland's assistance, it was Russia who pretty much did the final sweep by taking most of the successor states soon after it fractured and finally annexed the Crimean Khanate hundreds of years later.
You could probably argue that this is a stretch, but it was not entirely unlikely that a strong warlord could have reunited the lesser hordes just like Genghis Khan united the Mongols and started his conquests. This is pretty much why this fact is relevant.
So, in violation of the 20 year rule, I will now quote a guy who said "You forgot Poland". Dismantling the Teutonic Order was most definitely a team effort: it took Poland and Lithuania a while to pull off, they were pretty much equal partners in that endaevour. The last Grand Master eventually turned Protestant and became the Duke of Prussia.
So, the takeaways