Your question is rather broad, and seems to ask, why do people utter "I promise" and "what value is there of a promise". This is a Q&A forum, so there's no short answer, however, we can touch on some regions of philosophy that give some insight.
There are already a number of psychological and sociological answers here that raise questions regarding trust, but I'm going to offer some leads on the development of a philosophical response. Obviously, promises are about trust, both the psychological aspect of motivations and actions, and the sociological aspect that underlies collective intentionality, characterizes eusocial behavior, and undergirds the building of social institution. Promises also may be seen as contributing to social construction, where realities are built and shared by individuals.
Philosophers do carry out analyses on trust (SEP). A proper philosophical response would require an analysis of various aspects of including pragmatics, which is the study of what sort of meaning derives from real-world context, meta-ethical and ethical analysis, and how various minds interact, which involves the philosophies of mind, psychology, and sociology. As this board is a Q&A format, let's just note a few starting points for further discussion.
In philosophy of language, pragmatics contains ideas known as performativity. From the article:
Performativity is the concept that language can function as a form of social action and have the effect of change.1 The concept has multiple applications in diverse fields such as anthropology, social and cultural geography, economics, gender studies (social construction of gender), law, linguistics, performance studies, history, management studies and philosophy.
When someone says "I promise...", they are performing, in the philosophical sense. This involves questions about direction of fit which explores intentionality (SEP) and relationship between the mind and the world, and vice versa, sometimes subsumed by questions regarding mental causation (SEP).
In ethics, one has to take a meta-ethical position, which is the act of choosing an theory. One such famous characterization is the deontological (SEP) which explores right choice and action through the lens of duty, and is exemplified by theories such as the categorical imperative. Thus "I promise" is understood as "I have a duty".
Lastly, one might look at how the words "I promise" and understand how exactly they build a shared consensus and further collective intentionality (SEP). Such an exploration has been undertaken by Searle and others, and explores exactly how societies build and share knowledge (SEP), which is the epistemological aspect, or agreement on what exists, which is an ontological aspect. If one uses psychology and sociology as a springboard for philosophical discourse, one is taking a naturalistic approach to social construction (SEP).
The philosophical analysis of trust is not as extensive as classical questions, such as free will, dualism, causation, and so on, but one can presume that there are academics out there now stretching the boundaries to include trust. A good place to start looking is PhilPapers.org.