At the core of Inuyasha—deeper than demons, shard-collecting, and saving the world—is a love story gone wrong. The entire plot is based on events that occurred fifty years prior to the main Inuyasha timeline, and given that the entire context of Inuyasha is born from the rift in Inuyasha and Kikyou's relationship, it only makes sense that rift must be mended for the story to conclude. Although the journey to conclusion is long, Inuyasha manages to provide a solid sense of closure to the relationship between the unfortunate lovers.
At the beginning of the series, it seems that Inuyasha bears intense hatred for Kikyou, but it's soon revealed that they were originally deeply in love. Fifty years before Kagome awakens Inuyasha from his slumber, Inuyasha and Kikyou planned to use the Shikon no Tama, a powerful wish-granting jewel, to be together. Instead, they ended up betraying each other; Inuyasha mortally wounded Kikyou to steal the jewel, but Kikyou stopped him by shooting him with an arrow before dying.
When Kikyou is brought back to life fifty years later, she remembers Inuyasha's betrayal, and the feelings between the two pick up where they left off—malice. Over the course of the series, however, it is revealed that neither of them actually betrayed the other; rather they were victims of an elaborate scheme devised by Onigumo (eventually Naraku) to separate them. As they find out the truth, the bitter feelings between the two are slowly replaced again by the love that they used to share.
The purpose of mending their relationship is not so that they can resume their love, however, but so that they can finally put the past to rest. The circumstances are different now; Kikyou is not even human anymore, and Inuyasha has a new love interest. The final kiss between Inuyasha and Kikyou is not one of current lovers. It's the first step in correcting the wrongs in the world. The bad blood between them led to Kikyou's tainted soul, which led to the Shikon no Tama's return to the world, which in turn led to the entire mess with Naraku. Now that Kikyou's soul has been saved and laid to rest, the next issue, Naraku, can start to be resolved.