The word "Aside" in brackets or parentheses indicates that the speaker of an indicated passage of dialogue is speaking his private thoughts aloud but that they are not being overheard by whoever else is present. It is a stage convention which audiences understand and accept, although it seems artificial today and is seldom used in modern plays.
An example of an "aside" in King Lear is found in Act 3, Scene 6, where Edgar is pretending (counterfeiting) to be a homeless lunatic:
[Aside] My tears begin to take his part so much,
They'll mar my counterfeiting.
These usually short snatches of dialogue are probably called "asides" because the actor is expected to turn his face aside and probably hold his open hand against the side of his mouth to indicate that he is not being overheard. An "aside" always means that the character is speaking a private thought aloud when one or more other characters are present.