Richard III ends like every other Shakespearean tragedy – there's some major bloodshed and our hero/protagonist goes down pretty hard during the Battle of Bosworth Field.
But even though the final act of the play is a "tragic" ending for Richard, it's a glorious new beginning for England – thanks to Richmond/King Henry VII, who defeats Richard in battle (Stage Direction, Act 5, Scene 8). The outcome of this battle, by the way, puts an end to the Wars of the Roses and kicks off the Tudor dynasty, so it's a pretty big deal and deserves a grand gesture (i.e., England's new king needs to deliver a big, fancy speech).
If you haven't read it already, check it out below before reading our take on it.
And then, as we have ta'en the sacrament,
We will unite the white rose and the red:
Smile heaven upon this fair conjunction,
That long have frown'd upon their enmity!
What traitor hears me, and says not amen?
England hath long been mad, and scarr'd herself;
The brother blindly shed the brother's blood,
The father rashly slaughter'd his own son,
The son, compell'd, been butcher to the sire:
All this divided York and Lancaster,
Divided in their dire division,
O, now, let Richmond and Elizabeth,
The true succeeders of each royal house,
By God's fair ordinance conjoin together!
And let their heirs, God, if thy will be so.
Enrich the time to come with smooth-faced peace,
With smiling plenty and fair prosperous days!
Abate the edge of traitors, gracious Lord,
That would reduce these bloody days again,
And make poor England weep in streams of blood!
Let them not live to taste this land's increase
That would with treason wound this fair land's peace!
Now civil wounds are stopp'd, peace lives again:
That she may long live here, God say amen! (5.8.3)
There's a whole lot to say about this passage, but here's what we think is most important: Richmond/Henry VII's final speech tells us that Shakespeare is interested in continuity and the restoration of political order. Here Henry (a Lancaster) promises to reunite the two warring royal households (York and Lancaster, "the white rose and the red") by marrying young Elizabeth, the daughter of the late Yorkist King Edward IV. Even though a whole lot of Yorkists and Lancastrians have been wiped out during the grisly civil wars, the play suggests that King Henry VII's reign will usher in a time of peace and unity, which is all part of God's will ("fair ordinance").
Also, did you pick up on the way Henry mentions his future "heirs"? That's Shakespeare's way of giving his monarch, Queen Elizabeth I, a major shout-out (she was the granddaughter of King Henry VII and his wife Elizabeth).
P.S. You probably noticed that Henry talks about "smooth-faced peace" as if "she" is human. Talking about something non-human as if it had human qualities is called "anthropomorphism." Richard does this at the play's very beginning when he tells us that "grim-visaged [grim-faced] war hath smoothed his wrinkled front" (1.1.1). Interesting that the play begins and ends with the same concept.