This rule pretty much means that, if you were to be given a ripped out piece of paper. Even tough you are not the owner of the death note, the people you write down in there will die.
In the anime we can see a good example, I don't know the exact episode(s) but it was during the investigation of Light by the FBI as potential suspect. Where light gave the FBI agent Penber a envelope.
Light tells Penber to take the contents out of the folder he gave him, and Penber takes out four envelopes with ten rectangular spaces cut out of the left side of each one. Light says to write down the names of the agents into the blocks on the folders. As Penber writes, Light tells him not to get off of the train when he's done, and that if he follows his instructions his fiance and family will not be killed. As Penber finishes, Light says to put the transmitter and envelope back into the folder and sit quietly for thirty minutes, then place the envelope above on the rack and leave the train when no one will notice. At 4:42 PM, Penber dies from a heart attack three seconds after he exits the train, and his last sight is of Light’s face. Penber appears shocked as he looks at Light's face and says "Light Yagami." Light watches as Penber dies, and says "Farewell, Raye Penber." - Wiki
Light takes the files home and takes the pages of the Death Note out of their envelopes, revealing the times and causes of death for each agent. Each agent will receive the file and die at a different time.