When Romeo discovers Juliet dead, he wants Balthasar to leave so that he can be alone with her—because he plans to kill himself.
Balthasar is Romeo’s man. He is directed to carry messages, and he accompanies Romeo. When they arrive at Juliet’s tomb, Balthasar realizes that Romeo wants to be alone. Romeo is grateful, saying he is a good friend. Balthasar is concerned though. Instead of going, he sticks around.
For all this same, I'll hide me hereabout.
His looks I fear, and his intents I doubt. (Act 5, Scene 3, 107)
Balthasar is pretty convinced that Romeo is about to do something he shouldn’t. He worries that Romeo will act rashly, killing himself out of grief for Juliet’s death.
Romeo is so caught up in Juliet that he cannot think straight. He gets poison from the apothecary and goes to Juliet’s tomb, where he has to fight and kill Paris in order to be alone with his wife’s corpse.
Sadly, Juliet is not really dead. When Friar Lawrence arrives, Balthasar does not want to go with him into the tomb.
My master knows not but I am gone hence,And fearfully did menace me with death
If I did stay to look on his intents. (Act 5, Scene 3, 110)
Soon after Romeo takes the poison, she wakes up and finds him dead. The friar comes in just as Juliet is waking, aware of what has happened. She cannot be comforted. In her grief, she kills herself.
Balthasar does not seem to have accomplished much by sticking around. He is mostly a messenger, explaining what has happened and what he could not prevent. Yet Balthasar’s story, as unsophisticated as it is (he says he dreamed that Romeo slew Tybalt), gives important support for Friar Lawrence’s testimony to the prince.
Sending his man away and threatening him with death demonstrates Romeo's resolve. He wants to make sure he is not interrupted for any reason. In his grief, he intends to kill himself and wants to make sure Balthasar does not stop him.