I must admit that I am a little bit biased in answering this question, because I study Anthropology, but I hope that you will agree with these reasons as to why I feel it is an important and valuable field of study.
There are several sub-fields of Anthropology, each bringing something special to the table. Linguistic Anthropology deals with language, Biological Anthropology focuses on evolution and the "hard science" of humans, and Social Anthropologists work with human culture as it is created and re-created. Some sub-fields, like Nutritional Anthropology, create a crossover between other fields, like Biological and Social. As a field of study, Anthropology has many goals, but above all we seek to understand who we are as a species and how we can use our knowledge to solve problems.
Let's consider Biological Anthropology for a moment. Someone who works in this field may be interested in the story of human evolution-- where did we come from? Tracing the path of evolution throughout time and across space helps us understand what environmental pressures triggered shifts in speciation. In the world of academia, studying the evolution of primates and uncovering how we came to be Homo sapiens helps us to understand our part in history and potentially the role we will play in future. In a more real-world, everyday life setting, Biological Anthropology can foster a greater sense of unity among humans. The imperialism of the 15th through 20th centuries was "supported" by a belief that people of color were biologically lesser than white Europeans. Biological Anthropology has proven such racist and ethnocentric attitudes to be untrue! We are all Homo sapiens, in our many shapes, sizes, and colors, so a biologically-backed racism just does not hold up.
The study of Social (or Cultural) Anthropology encourages greater communication, cooperation, and respect among humans. As we grow from infants to adults, we learn a shared set of cultural symbols and practices from the people around us. Sometimes, we encounter cultures and individuals that have very different sets of symbols and behaviors. While it can be surprising or upsetting at first to find that we have differences from one culture to another, Social Anthropology seeks to uncover the common motivations behind our lifeways. For example, Marvin Harris wrote a rather famous article about the Sacred Cattle of India. As an American, I have grown up learning that cows are animals raised to be eaten-- so why would cows be offered a sacred status which forbids killing them? Historically and even today, cows are of more value to the people of India alive (as a source of fuel, milk, and labor) than dead (as a few meals of beef.) At the root of both of these attitudes on whether or not cows can be killed to be eaten is a desire for survival.
Nutritional Anthropology is an exciting field of study that can answer some very troubling questions we face as a species. In a world where we produce more than enough food to feed all living people, why do so many go hungry? How do we fix this problem? Nutritional Anthropologists study not just the quality and quantities of foods but the systems that govern how food is produced and distributed. Some Nutritional Anthropologists have been involved in the development of high-nutrition, low-spoilage food products that can be used to combat hunger in impoverished places.
I think that in every Anthropologist is a desire to answer the question, "How can we be better humans?" An admirable pursuit, indeed.