The excitement generated by the successful completion of studying the entire Torah Shebichsav - as individuals preparing the text with targum and meforshim and as a community listening to the leining and participating in public study - has its roots in several sources. The Ramban in his commentary to Parshas Mishpatim (24:11) relates our annual dancing and perhaps any siyum to the recorded celebration of our leaders upon receiving the Torah at Sinai. He writes, "... and they [the nobility] ate and drank - they celebrated and made a Yom Tov for one is obligated to celebrate the receiving of Torah." The Ramban parallels these occasions to the feast that Shlomo Hamelech feted in Yerushalayim after being granted unsurpassed intellectual gifts (Divrei Hayamim 2, 21:12) and to Dovid Hamelech's great feast marking the preparations to establish the first Beis Hamikdosh (Divrei Hayamim 1, 29:21).
At first blush these events and seeming precedents for our Simchas Torah are not similar at all. The festivities of Matan Torah and of Dovid and Shlomo HaMelech celebrated the anticipation of horizons of spirituality and knowledge that had just been unlocked and laid out for them. They correctly perceived that they had been catapulted beyond the boundaries that hemmed in the most talented people of all times, and what a passionate simcha they must have experienced. Isn't the annual completion of V'zos Habrocho quite different? Are we not expressing our happiness and gratitude for the privilege of accomplishing the understanding of part of Hashem's instruction?
Perhaps the Ramban wants us to understand that every siyum, be it of a mesechta or the entire Torah Shebichsav, is celebrating not the accomplishment but rather the anticipation of applying all of one's knowledge to future studies and situations. Indeed that is why at every siyum we focus on the "Hadrans", praying and promising that we will return to the mesechata at hand and that the mesechta has become an active and alive part of our consciousness. Thus the simcha has its roots in kabolas Hatorah and the joy of Shlomo Hamelech upon his receiving the gates of all knowledge.
In similar fashion we can appreciate the explanation offered by the Avudraham for our rush to start Breishis as soon as we have completed Sefer Devarim. He refers us to a challenge that the Satan throws in an attempt to deride Klal Yisrael. He claims that now that we have completed the study of the entire Torah we will be putting it away and presumably go on to other masterpieces, lehavdil. Upon hearing the beginning of Bereishis immediately after chazak chazak, Hashem's confidence in us and our knowledge of the absolute singularity of Torah and its place in our lives, is vindicated. Perhaps the medrash is also pointing out that Klal YIsrael sees the successful completion of one cycle not as an end but rather as a new rung in the ladder on which to penetrate the texts ever so more deeply on the next way around. Our simcha on this Yom Tov celebrates our well founded expectation that we will always find new insights in the Torah and the blessing that we have as Torah and life continuously illuminate each other.