It makes sense without the [...] if the sentence is in its original context, but when they take it out of the context, they have to supply a subject or referent in brackets so that the sentence makes sense by iself.

Essentially, it represents the current writer’s modification, however slight, of the actual quoted text. It’s a mark of care, to show that “the writer didn’t actually say this, but he implied it”, for example, or “I’m cutting part of what was said in a way that doesn’t damage the context or meaning, but isn’t relevant to my purpose in making the quote in the first place.”

Much easier to do that with ”[ ]” rather than tediously stating what I said above.

Adding to what others have said, it’s often used to change personal pronouns or possessive adjectives into proper nouns. For example, a line might originally read “Clinton remarked ‘He has no sense of humor’”. This causes you to think “who, who has no sense of humor”. So the author changes it to “Clinton remarked ’[Obama] has no sense of humor’”. (Totally made that up, don’t read anything into it). They can do this if the conversation has already established that the interviewer and Clinton are talking about Obama, but it was just 10 minutes of stuff that won’t make it to print because it’s boring or irrelevant or not fitting with the narrative or whatever.
More often, when they cut out a part, they use an ellipses (...) instead .

What they said. If my friend says “I love beets,” it looks better for me to write “My friend says ’[she loves] beets’” than “My friend said, ‘I love beets.’” One clause instead of two is cleaner!