Instruction
Lewis writes this book in order to dispel a common assumption that the English word "love" holds only one meaning. He writes to instruct the reader upon the multiplicitious and complex nature of love, rooting his arguments in the meanings of the four Greek words for love. In this way he introduces the reader to the importance of specificity of language, preventing ambiguity and insincerity. More importantly he's instructing the reader as to the varied nature of love itself. From affection to total, divine love, the four definitions serve to clarify the holistic and radical potential of love. Most importantly, Lewis teaches the reader to recognize how each of the four loves manifest in their lives and how they may manifest them intentionally.
Selflessness
Through this discussion of the various meanings of love, Lewis expresses the desire that selfless love abound in the world. He desires the reader to recognize the value of selflessness as practiced through love. This process is radical in the sense that Lewis relates his instruction to obligation. By receiving this information, he argues, the reader must necessarily participate in the four loves. He presents obligation through the framework of selflessness, presenting the benefits of living a selfless life. Most importantly, he presents a series of levels of participation. This includes everyone, explaining how every person demonstrates at least one of the four loves. By recognizing the ways in which one already participates in love -- giving or receiving, -- the individual is able to identify the benefits of attempting all four.
Christian tradition
In an appeal to Greek tradition, Lewis is arguing for the historical interpretation of biblical scripture. He's writing to a primarily Christian audience, hoping to dispel assumptions. He uses Greek to make the Bible's original text more clear because the nuance of the four words for "love" has been lost to the single English equivalent. Thus Lewis does not necessarily present any original thought in this book, but he clearly communicates the obscure truth of the original text in order to equip his readers to further interpret the text on their own. In a sense, he does not want to persuade the reader to do anything but read the Bible for themselves and to continue to refer to the most accurate historical understanding of the text as possible.
