DR is a ranking system that you can use as a basis to know how fast you are compared to other drivers. An E rating means you haven't played enough Sport mode yet. Get 5 races in and you'll be promoted to D, the lowest rank. As you get better you go up through the ranks C, B, A, and finally, A+. However, you cannot always rank up, as you might hit a certain skill cap where the drivers you are matched against are always faster than you. That is called having a plateau, and in a plateau you'll find that your race results would keep your DR in roughly the same place. If you hit a plateau, it means you've reached the DR ranking that your skill level is capable of

You can improve though, since in a plateau you'll be constantly matched with faster players, there's lots of opportunities to learn
Your DR rating goes hand-in-hand with your SR rating. SR is basically how clean and well-mannered you drive. The ranking goes from E all the way to A, and then S (although you start with SR B). You cannot progress through your DR rating if your SR rating is too low. If your DR is high enough, and your SR rating gets low enough, you'll get a DR reset. A DR reset means that the game resets you to a lower DR rating because you lost too much SR. As a rule of thumb, getting to SR S as fast as possible is one of your primary goals, as the racing is cleanest in that ranking, and you have the lowest chances of getting a reset