Why does putting your name on a health behavior contract indicate your commitment to the contract?

Because it is written. A popular tool used for motivation and commitment is the use of written contracts and often does not even need to be as lengthy as a contract. A slip of paper or even another writing surface has been shown to significantly increase the likelihood of committing to that action. 

There is also premeditation involved when signing a contract or writing on a piece of paper. An individual has read and recognized the written words and then signed it according to a commitment to that act. As such there is a significant amount of thought put into the over procedure, and thus increases again the likelihood of an individual to be committed to completing those actions.

Aside from all the factors of motivation, the signing demonstrates commitment to the contract because that is fairly universally understood to be the way in which people recognize and formally commit to contracts. Whatever the context of the action, whether signing papers to buy a house or for a new job, the signing of the contract is the commitment of the individuals involved to complete the requirements within the contract. Although in this instance the healthier behavior most directly affects the individual who signs the paper.