A dagger or sword or any metal weapon, if not maintained, will rust. In Shakespeare's day, sheaths were leather, and if the metal weapon was left inside unused or unmaintained long enough, the acids from the leather would corrode, or "rust" it. Similarly, if blood was allowed to remain on the metal after stabbing, and not wiped off, that would etch or "rust" the metal, leaving hemoglobin stains. Shakespeare employs a brilliant image in implying that the dagger will never be removed from its sheath, or Juliet's bosom, and therefore will rust; the word also suggesting the eternal decay as is previously mentioned.
It means she's going to kill herself. Juliet's chest becomes the dagger's sheath (place you store your cutlery). "There rust and let me die" refers to the eternity of rotting death they will spend together. She calls it happy dagger somewhat ironically, since it's giving her what she wants, but it is still killing her.
When Juliet wakes up she sees that Romeo has drunk the poison. So she immediately takes his dagger and stabs herself to death: "O happy dagger,/This is thy sheath.There rust and let me die." Juliet hears the noise of others coming and she quickly stabs herself without making a long speech.
She puns on the word 'happy.'Happy' could mean,
1. She was fortunate in quickly discovering the instrument to end her life.
2. The dagger is always kept sheathed in a metallic case, but now it is fortunate because it is sheathed in her youthful body.
3. The line also has an obvious sexual connotation.
"Romeo and Juliet" belongs to the early phase of Shakespeare's dramatic career - a phase which is characterised by excessive punning.
I confess that I am no expert, but playing Lord Capulet recently (for those not familiar, Lord C. is Juliet's father) with dreams of directing one day, I took a quite some interest in this and other notable lines. My own theory is this:
Juliet's descent to suicide begins with her discovery that Romeo's lips are warm; she has missed him by mere minutes. This is Fate's ultimate blow in a long series of blows. Immediately she hears the watch, signifying the return of life and the world as usual, though without him, but worse, a return to those whose obstinate insistence on perpetuating the feud put them there. I believe her grief turns to rage, something most productions I've seen seem to miss. (Olivia Hussey captures this very well in Zeffirelli's movie.) She would as soon lash out at those whom she sees as the real villains, but lacking that option, she directs the rage in the only direction she can--to herself, intent on punishing them through the discovery of her in death. This is consistent with the nature of suicidality.
Rust is the death of the metal, and an undistinguished one. In her state of mind she may choose to see it as even ignominious. Befriending this instrument of death, she pronounces such a death for herself, thus vicariously, for those she feels truly deserve their own.
I invite any comments on this theory.
(For me, this is unquestionably the most heart-rending line in all of theatre, by which I include print and film.)