Why is the topic of justice such an important one in philosophy?How would you come come to grips with a problem of justice?
Several reasons. Because philosophy is the basis for all forms of government, and because justice is a natural human desire, so it is therefore manifested in that philosophy.
The desire to change government or society often arises out of real or perceived injustices. Humans have a natural desire for fairness, though we often don't observe that. It is because of this desire, the philosophical changes that follow, and the societal and governmental actions that result that humans make progress in terms of justice over time.
As far as "coming to grips" with a problem of justice, it is essential to first acknowledge and recognize that an injustice exists, and to understand what its root causes are. Philosophy helps us to understand those causes, the natural precursor to any positive action towards justice.
Philosophers engage in a discussion about justice for a variety of reasons. The first is that it is entirely difficult to create a totalizing definition of justice that eliminates further discussion. Justice can be defined as fairness, as minimal government intervention, as the idea that individuals should be allowed to pursue self interest, as well as other conceptions. With this discussion, philosophers can examine and debate concepts and their implications within the realm of justice. This divergence nature helps to make the topic one worthy of discussion as well as an essential one because the very nature of society and government is determined through an understanding of justice and how it is to be implemented.
I would say that this topic is so important because it is a question that is fundamental to human existence. We all want to live in ways that are just (or at least almost all of us do) and in a just society. So then the question becomes "what is just, or what is justice?"
Philosophy is concerned with trying to answer questions like this that are central to human existence.
As far as how to come to terms with it, that is a question for 90 books, not 90 words. This question is what philosophers try to figure out.