If you understood the meaning of a repetition of a non-matching bracket expression (such as [^[]* which matches zero or more occurrences of any character except [), I'm surprised that you didn't understand the meaning of the matching bracket expression [[] which matches one occurrence of the [ character. Similarly, [^]] matches any character other than ] and []] matches a ].

You have to use the bracket expression [[] or escape the opening bracket \[ to distinguish it as a character to be matched (rather than the start of a bracket expression). In some contexts, you do not need to use a bracket expression or an escape to specify a closing bracket, but the meaning is is the same if you use []] and it is symmetric with the [[] if you have an editor that pairs up opening and closing parentheses, braces, and brackets.