Know Your LotRO Lore: The Nine for Men
For Men, wearing a Ring of Power caused invisibility and an extended lifespan, although not exactly how many had imagined. If worn too long, the Men would eventually become wraiths under Sauron's complete control. As an extension of this topic, you can look for next week's Know Your LotRO Lore to focus on these wraiths, better known as the Nazgul.
How was the invisibility explained exactly? For mortals (Men and Hobbits), the material plane was the only visible plane, while most Elves (with the exception of the Mirkwood Elves and few others) could see in both the material and spiritual planes simultaneously. When a mortal would wear a ring, they would appear invisible to other mortals (except Tom Bombadil, curiously enough), yet they were actually slipping into the spiritual plane. This is why the Elven and Dwarven Rings were not given that power of "invisibility."
The Nine Rings worn by the Nazgul were the only rings to be fully under Sauron's control and work as he had planned throughout their existence. Once they converted the kings of Númenor to Sauron's service, they remained together throughout the destruction of Dúnedain and the rise of the kingdom of Angmar.