One definition of commitment is "a pledge or promise, an obligation." When we promise someone that we will do something, that person is counting on us to keep our promise. If someone makes a promise to you, you depend on them to keep that promise. They expect the same from you.
Trust, knowing that someone will keep their promise, is the foundation of good relationships. That trust is built by keeping promises.
I'm not sure if I fully understand every bit of your question, so I'll try my hand at responding to the opening part: "Why do we keep commitments?" As I see it, we have at least three reasons for keeping commitments:
1. We want other people to keep their commitments to us, so we do the same to them. For example, if I repeatedly break commitments that I have made and word gets around (it usually does!), I can expect that people will be less likely to honor their commitments to me.
2. We feel some sort of external force (such as pressure or encouragement) to do what we said that we would do. For example, a man in a committed relationship may work extra hard to remain loyal and make the relationship work because he would be embarassed to have others see him fail and/or has positive role models and other supportive people who encourage him to work out any conflicts in the relationship as they arise.
3. We feel some sort of internal force (such as joy or shame) to do what we said that we would do. Item 3 may simply be the internalization of item 2, or it may be much more than that. In either case, I know that I feel good when I follow through on my commitments and do not feel good when I fail to do that same.
You may find it helpful to look up the terms "extrinsic motivation" and "intrinsic motivation."