Hello! I agree with Sandi! "Right" is used for emphasis in the phrase "kept right on". "Kept right on" means continuing to do something, even though the person is expected to stop. Example: I saw him on his bicycle. I waved to him, but he kept right on going! (He didn't stop to wave back) I told her to stop chopping tomatoes, because we had enough for the salad. But she kept right on going! (She continued chopping tomatoes even though she was supposed to stop) The meeting was supposed to end at 5 pm, but the boss kept right on talking until 5:30! (The boss was supposed to stop talking at the end of the meeting, but he or she did not) Keep in mind that the phrase "kept right on" is informal and has a mild negative connotation. Native speakers generally use this phrase with friends or family when they want to express confusion or frustration about someone's actions.

Right has no particular meaning on its own here. The phrase, in this case, 'to keep right on' means to continue or persist in (doing something). The "right" could be used to emphasize how he kept on talking without pause or consideration.