The last chapter of the first book is called "The Breaking of the Fellowship." But we've just met the Fellowship; why do they have to break up now? There are some pretty obvious plot-based reasons for the Fellowship to break up: for one thing, although they are all heading south up to a certain point, they have reached a fork in the road where some have to go east to Mordor (that's you, Frodo) and some want to go south to Gondor (ahem, Boromir). Frodo's departure also leaves Aragorn, Legolas, and Gimli free to ride to Merry and Pippin's rescue. And Merry and Pippin have been carried off by Orcs at just the right time to meet the Ents and sack Isengard down the road. The Fellowship of the Ring ends as all good first books in a series do: by laying out all of the chess pieces in the right places so that they can carry out their different roles in the next two books.
The ending of the novel also says a lot about the book’s moral purpose for Tolkien. The author has described the purpose of the Lord of the Rings trilogy as "the ennoblement of the ignoble" (Source, 220; in a letter to the Houghton Mifflin Co). In other words, he wants to show humble people doing great things. By starting The Fellowship of the Ring with so much back story about the Hobbits (the humblest of humble people, content with food, beer, jokes, and songs), Tolkien sets up an unlikely foundation for a much broader story of Good vs. Evil among all the peoples of Middle-earth. By making the Hobbits the heroes of the Ring quest, Tolkien conveys that even ordinary folk can rise to amazing things if they have to.
But what does all of this have to do with the ending of The Fellowship of the Ring? Well, now is the time for Frodo and Sam to shine on their own – two hobbits vs. the world – without the crutch of relying on "the high and noble" members of the Fellowship.