No amount of grieving can bring the dead back to life, so forget about her quickly and marry me instead.
When you sigh, you feel a certain type of emotion. A type that makes you feel sad and causes you to lament and to grieve.
For the reading, just think about this feeling you're having (this lamenting grieving one). Then, decide you're done with it. Throw (投げる) it out the window. You don't want these feels anymore!