Why study Greek instead of Latin? At least Latin is used (or used to be used) in science and law, among other areas.
It is important not to underestimate the importance of the Greek language in the history of Western thought. True, Latin played a large role in the development of taxonomic systems in areas such as law, medicine, and science. Latin was also the written language of intellectuals well into the early modern period. However, Latin culture (including literature, philosophy, science, and so on) was born out of Greek culture. The ancient Greeks came first. The ancient Greeks produced the first scientists, lawyers, and educators; and while the Roman Empire left a larger military footprint on the world, Latin ideas remain significantly indebted to their Greek counterparts. Also, there are many great cultural watermarks written in Greek. Homer's famous epics were written in an early dialectic of ancient Greek, and the New Testament was written in a vulgar dialect of ancient Greek that became popular after Alexander the Great ruled a significant portion of the world. Finally, the study of Greek does not preclude the study of Latin. In fact, these two languages for many years paired in the curricula of primary schools.
What is an inflected language? I don't understand why Latin has so many endings.
Inflection is an important feature of ancient languages. If a language is inflected, then the endings of various words will change in order to reflect their role in a given sentence. Although English is not nearly as inflected as its Indo-European predecessors, there are still traces of inflection to be found. For example, we distinguish between uses of 'I' and 'me' depending on whether you are referring to the subject of a sentence or its an object of some verb. Another ubiquitous example of inflection is the word 'hence', which originally meant 'from here'. The idea of coming 'from' somewhere was built into the very form of the word 'hence', and so a single word could contain a great deal more information. This is exactly the case in Latin, where almost every kind of word (verbs, nouns, participles, etc.) are inflected. Although tedious to memorize, these endings give readers a great deal of information about the role each word is playing the sentence. So much information, in fact, that word order is hardly relevant in Latin (unlike English), since the grammatical roles of each word can be gleaned from the endings alone. In English, in contrast, word order is the most important feature of any coherent and well-formed sentence. In the end, Latin's inflection gave speakers and writers a great deal of creative space. Learning the endings is well worth the effort.
Why should anyone bother to study ancient philosophical texts that were written over 2000 years ago?
It is important to remember that the value of ancient philosophical texts, such as those written by Plato and Aristotle in the 4th century BCE, is not solely a function of the things they got right--although we may very well have much to learn from them! These texts tell us a story about how we have come to ask the very questions we ask today. Given any philosophical question we might ponder today, there is always a corresponding question or insight first explored by the ancients. Sometimes these questions remain unchanged. For example, we are arguably no closer to answering the question 'What is the best kind of life for a human to live?' than we were 2,000 years ago. Even in areas where tangible progress has been made, such as in the empirical sciences, we often benefit from reflecting on the origins of our scientific modes of thinking and the earliest attempts of making sense of the natural world. While our knowledge of the physical constituents of the world has surely advanced, deep philosophical questions remain even for our best scientific theories. These questions look awfully similar to those asked by our predecessors. 'How did something (e.g. the world) come from nothing?', 'How can finite creatures grasp the concept of infinity'? On a final note, it is worth mentioning that even in cases where our philosophical predecessors went deeply astray, they remain a fruitful starting point for inquiry. The texts they have written serve as partners in the dialogue that constitutes the history of ideas.