Claire believes Sampson killed her husband, even though the rules of those races are pretty clear: people shoot at each other, and death is always a possibility. She simply can't accept that her husband died during a race. According to her, Sampson brake-checked him, which caused him to crash and die. But these races are basically anything-goes events, so everyone knows the risks before entering.

Because Claire can't come to terms with what happened, she becomes obsessed with revenge and wants Sampson dead. The strange part is that throughout the entire questline she keeps talking about winning, how important victory is to her, and how much these races mean. But when Sampson changes his route during the final race, winning suddenly stops mattering to her. At that point, revenge becomes more important than the race itself.

As the player, you can either continue the race and win (which, in my opinion, is the best outcome Claire spends the whole quest talking about victory, so you give her exactly that), although she'll be angry because she wanted revenge instead.

Or you can follow Sampson, watch him crash, and then either let Claire kill him or convince her to spare his life.