Part of the answer to this has to do with thinking about the assumptions underlying the question. The phrasing suggests that you consider prose the "unmarked" or "normal" type of language for drama and Shakespeare's use of verse somehow distinctive. This, however, was not the case. Western drama originated in ancient Greece and Greek drama was always composed in verse, as were the subsequent Latin dramas. When secular drama was revived in the Renaissance, it was influenced by classical models and written in verse. Thus for audiences of Shakespeare's period, the effect would have simply been that Richard II was a typical drama. As one innovation of the period was making some lower class characters in plays speak in prose, the predominantly royal or aristocratic characters that predominate in this play would have been expected to speak in verse.
What prophecies and curses do within the play is make Richard II's downfall appear inevitable. As Shakespeare's period was one in which people presumed the "divine right" of monarchs, with monarchical authority being legitimized by God, overthrow of a monarch can appear aligned with forces of chaos, undermining civic, natural, and divine orders. What the prophecies show is that the overthrow of Richard II is part of the divine order rather than a rebellion against it. E. M. W. Tillyard's short book, The Elizabethan World Picture, is an excellent resource for understanding this type of thinking.
When Shakespeare and other playwrights of the same period wrote in verse, they elevated the tone and form of the plays to convey a more poetic and sophisticated milieu. It is often said that one can tell which characters are of the noble classes and which are of the lower classes in Shakespeare's plays, because the former speak in verse and the latter speak in prose.
Curses and prophecies serve to emphasize the political impact of Richard's reign and the effect his actions will have upon the future of the kingdom. There is a sense that his reign is cursed by his own arrogance and impulsive actions. This speech by Gaunt indicates a sense of Richard's reign being cursed down the ages:
. . . Thy deathbed is no lesser than thy land,
Wherein thou liest in reputation sick;
. . . A thousand flatterers sit within thy crown,
Whose compass is no bigger then thy head . . .
O, had thy grandsire with a prophet’s eye
Seen how his son’s son should destroy his sons,
From forth thy reach he would have laid thy shame,
Deposing thee before thou wert possess’d. . . .(II.i.93-107)
The speaker suggests Richard's flaws are connected to his choices and actions, as opposed to being passed down through his ancestral line or encouraged by family tradition.