In medias res is Latin for "in the middle of things." A story that begins in medias res typically starts off with a bang, so to speak. The audience is thrown into the action right away with the characters, rather than slowly building to the action.
Richard III begins with the titular character laying out the context of the setting in a famous soliloquy: he says who is in power, how badly he wants that power for himself, and how far he's willing to go to get what he wants. Right after, Richard puts his evil plans into motion almost immediately.
That Richard III opens in this immediate way makes sense. Richard III is technically a sequel, following the plot of the Henry VI plays. By this point, the House of York has claimed the throne after deposing Henry VI, and they are celebrating their victory, unaware that one of their own plans on carrying out such nefarious schemes.
By introducing the cunning, crafty Richard in the midst of these joyful proceedings, already underway in his plans to seize the crown for himself, Shakespeare is already setting up an atmosphere of suspenseful intrigue. He gives the audience no time to settle comfortably into the world of the story. They are on the edge of their seats from the beginning.