How is it believed that the Four Noble Truths lead to happiness?
I think that the Buddhist might have to orient the question a little bit. I don't think that the Four Noble Truths "lead" to happiness. I don't see Buddhism in such a light of direct cause and result. Rather, I sense that the Buddhist would say that the Four Noble Truths leads to greater understanding and awareness of this life and the consciousness in which one finds themselves. It is here where understanding can be associated with a linking of oneself to a universal construct, in which happiness is already understood. It is a minor point, but I believe it is one that needs to be brought out.
For the Buddhist conception of the Four Noble Truths, they begin with an understanding or acceptance of dukkha, or pain in being. This is the condition of anxiety or a constant sense of changing that is associated with this life, one of samsara. For the Buddhist, it is here where all other truths emanate, as the Lord Buddha himself sought to understand why theire is pain, suffering, and a sense of constant change in being in the world. Through the conception that one's being, the sense of dukkha governs all being in the world, the second noble truth is to understand the condition of dukkha. This is caused by craving and by coveting that which causes frustration and helps to contribute to the state of being in which there is constant change and a lack of totality. In understanding the craving, yearning, and desire for attachment that causes dukkha, Buddhists believe one can recognize their own place in the world and perhaps construct their own reality as a way to overcome in a transcendental manner such a notion of being. The cessation of dukkha is the third Noble Truth and it essentially calls for individuals to view what is as merely that, what is. As opposed to being attached to what should be, what should not be, or what can be, individuals who understand dukkha and accept reality for what is cease it from causing pain to them. The Fourth Noble Truth involves this in terms of being able to live a life where cessation of dukkha is evident on a daily basis with the advent of right living. In these four truths, the Nobility of human beings in modeling themselves in the light of Buddhist understanding is evident, a four pronged approach to understand what is the samsara condition that engulfs all being. It is here where the Buddhists believe individuals should concentrate their efforts as it links the individual to something larger than their own understanding of self.