Confused about the correct use of brackets in English writing? Don't worry, you're not the only one. The good news is that there are limited uses for brackets in most citation or formatting styles, so there aren't very many rules to learn.
The most common use of brackets in academic writing and journalism is to add editorial content that is not otherwise part of the original quote. Writers and editors do this for several reasons, including:
Example: The speaker noted, "That year  was the year we saw success."
In the above example, the writer has added  to clarify the year to which the speaker was referring. This was done because the audience reading the quote would not otherwise know this information, usually because it was not included in the context of the quote (but elsewhere in a speech or piece of writing).
Example: He looked deeply into her eyes and said, "je t'aime [I love you]."
In the above example, the speaker didn't speak the words in English. This was added by the writer or editor to translate words that the audience might not know the meaning of.
To indicate a change in capitalization
Example: "[J]ust don't text and drive," said the teacher.
In the above example, the quote was taken from the middle of a sentence, "I don't care how much you're on your phone when you're at home, but just don't text and drive." Since the writer or editor only wished to use part of the quote, the small case "j" was changed to a capital letter, so brackets were needed for it.
To indicate an error
When quoting someone who makes an error in their speech or written words, brackets are often used around the word sic.
Example: "He were [sic] a thief yesterday," said the store owner, when filing a police report."
To note added emphasis
A writer will often add visual emphasis to a word to get the reader to focus on it and when this happens, those changes should be noted in brackets.
Example: "Our world is a much [emphasis added] different now that social media has taken over," she said.
To note objectionable content that has been removed
When a writer chooses to intentionally leave out objectionable content from a quote, brackets are used to note that removal.
Example: "Get the [expletive] out of here now," he shouted.
To separate levels of parenthetical parts of the sentence
This is not something you will encounter often, but when it happens, brackets can be used to create another level of parentheses in a sentence.
Example: In his research on the mating habits of the shark, Bedford noted the role that environment must play in ensuring a healthy population (see Luther et. al  for the details of this research).