Richard isn't just another run-of-the-mill protagonist (a.k.a. main character). The guy pretty much dominates the entire play in a way that no other Shakespearean character had done before. (Remember, Shakespeare hadn't yet written Macbeth or Hamlet.)
Because Richard is always addressing us (the audience) directly and basically acts like our tour guide for most of the play, we can't help but identify with the guy, even though he is most definitely a villain. This is why Richard is so often referred to as an antihero – a protagonist who lacks hero qualities like strong morals, honesty, courage, and strength. (Richard doesn't win the crown through any heroic feat – he basically lies, cheats, and murders his way to the throne.)
This is a play about rooting for the bad guys – at least until the end, when the good guys show up and we're obliged to switch sides. It's sort of like watching Ocean's 11 and hoping the thieves get away with their crime. Richard may be a twisted kind of antihero, but he's the guy responsible for most of the action, and he guides us through it, so we can't help but like him.
When talking about Richard as a protagonist, we said we root for the bad guy until the very end. Well, when Richmond (a.k.a. King Henry VII) shows up and defeats Richard during the Battle at Bosworth Field in Act 5, he becomes England's new king and our instant protagonist. Ta-da! Time to switch sides, because it's King Henry VII's new reign that puts an end to Richard's villainy and, according to the play, brings about peace and prosperity in England.