From the really small amount of Latin & Italian I know (Very small amount, mostly from youtube educational videos - not italian/latin focused) i know that monte means mountain and negro is black.
In Greek, my mother tongue, its called "Μαυροβούνιο" which means exactly the same thing.
So how did it get its name?
For this we have to go back to the middle ages. In the 13 century the region was called Zeta,after the river that flows in the nearby valley.Near the town of Kotor (owned by Venice) was a mountain called Lovćen which (then at least) was really wooded area,so the region around the mountain was called Crna Gora (meaning 'Black Mountain'). As the Ottomans invaded the country shrank to the mountain Lovćen and the country got called Crna Gora. The name 'Montenegro' is the Italian translation of the name. The Turks have their own name for us (I belive Karadag) and the Germans used the German version until the 20th century and WWI.