Brinksmanship was caused by the fact that the US and the Soviet Union both felt that the Cold War was so extremely important. Neither side wanted to look weak in the face of the other because neither side wanted to let the other have an advantage that could eventually lead to it winning the Cold War.
For both sides, the stakes in this conflict were extremely high. It was (they felt) a life-or-death struggle. To the US, for example, letting the Soviets win would mean that freedom would be wiped out in every corner of the globe. If winning the conflict is that important, then of course you engage in brinksmanship. You take things as far as you can (as seen most clearly in the Cuban missile crisis) before you back down.