Philosophically, the siyyum is a way to celebrate your accomplishment with your community. Especially since the siyyum requires (1) a minyan, so you can say the kaddish derabbanan, and (2) a celebratory meal, it's a way of sharing your personal study accomplishment.
The hadran is one's declaration of intent to return to this subject matter again someday. Just as when we finish reading the book of Devarim on Simchat Torah, we immediately begin again with Bereshit, the idea is that we are completing a phase of study but not, God forbid, completing our study of Talmud without the intent to return.
When I celebrated my fortieth birthday, I timed my completion of learning a masechet of Talmud to make my birthday party be more than a secular-style party; it was a religious celebration as well.
As others have said, the minimum amount of text that justifies a siyyum is dependent on the person's abilities and experience. In my community, the general standard is a masechet (tractate) of Gemara or an entire seder (order) of Mishna; but for those who are first coming to the study of these texts, learning in depth one's first masechet of Mishna is worthy of a siyyum.