Kusha is grass , also called as Darbha .
When Amrit was obtained from samudra manthan , few drops of it fell on Kusha grass , while devas and asuras were fighting over it. Actually Kusha grass is chest hair of vishnu. When mountain mandar was supoorted by vishnu avatar koorma/leather back turtle, the friction produced by rubbing mountain over the back of koorma produced Durva/ Darbha/ Kusha grass , hence its pious and is used for religious purposes because its pure.
Princes who are to be anointed are asked to fast and sleep over mat of kusha grass , that is to signify that they are not there to rule but there to serve people
In VR , Ram has to fast ( maybe Sita too) Vashishtha does a homa and gives Ram a blob of clarified butter /ghee and he has to fast that night and sleep on kusha grass ..mentally being prepared to face challenges in a king's life
Alternatively read :
http://www.sanskritimagazine.com/indian-religions/hinduism/significance-of-darbha-or-kusha-grass/