Every Yom Tov has two aspects: On the one hand, it is a day of tefillah and study; a day on which we spent many hours in shul engaging with ה'. Indeed, the word מועד means a time of meeting - a time when we rendevous with הקב"ה. On the other hand, Yom Tov is also a day of celebration, when we eat festive meals with friends and family; in fact, the משך חכמה suggests that the permit to cook and carry and perform other food-related melachos on Yom Tov is given by the Torah in order to promote such celebration.
The Gemara in Pesachim finds a source for these two aspects of Yom Tov in the verses: עצרת תהיה לכם and עצרת לה' אלקיך; Yom Tov has both an aspect of לכם - for you - a day of human, physical celebration, and an aspect of לה' - for Hashem - a day of spiritual activity.
The Gemara there records a dispute between ר' אליעזר and ר' יהושע regarding the relationship between these two aspects. According to ר' יהושע Yom Tov should be divided equally between these two aspects: חציו לה' וחציו לכם. (The halacha follows this view.) According to ר' אליעזר, however, one has a choice: או כולו לה' או כולו לכם; one can devote the day entirely to ה', and forgo eating altogether, and devote it entirely לכם.
However, continues the Gemara, even ר' אליעזר concedes that on Shavuos there must be an element of לכם along with לה'. Why? Because it is יום שנתנה בו תורה, the day on which the Torah was given.
This requires explanation - what connection is there between the fact that the Torah was given on Shavuos and the fact that the celebration of Shavuos must be divided between these two aspects, even according to R' Eliezer who does not require this division for other Yomim Tovim? The commentators, of course, offers various explanations; however, I would like to offer an explanation of my own.
The Gemara in Shabbos tells us that Yom Tov is an אות, a sign of the covenant between הקב"ה and the Jewish People. (For that reason, the Gemara explains, we don't wear tefillin on Yom Tov; because tefillin are also such an אות, and it is redundant to have two אותות.) I would suggest that it is because Yom Tov is an אות that its celebration requires the elements of לה' and לכם, both the physical and the spiritual, the human and the divine, since Yom Tov is a sign of the ברית that connects them.
Both ר' יהושע and ר' אליעזר agree, I would suggest, that these two elements must be present in every Yom Tov. But while ר' יהושע concludes therefore that each person must observe both לה' and לכם, dividing his observance of the day between these two aspects, ר' אליעזר maintains these celebration can be divided among different people, with some members of the nation focusing on the לכם aspect, while others focus on the לה' aspect, so that together the entire Jewish nation construct a full Yom Tov celebration.
However, this is true on the other Yomim Tovim - Pesach and Sukkos - whose theme are essentially national in character. On Pesach we celebrate our freedom as a nation, and on Sukkos the ענני הכבוד which represent the special providence with which הקב"ה guides our national history. But Shavuos is different. Because Shavuos is יום שנתנה בו תורה, the day on which the Torah was given.And while the Torah was offered to the entire people, it has to be acquired by each individual on his or her own. The study of Torah is essentially an individual challenge; and what we celebrate on Shavuos is not only the Torah that was given to us, but also the Torah that was given to me. The Gemara itself makes this point when it tells of R' Yosse who would punctuate his celebration of Shavuos with the comment: אי לאו ההוא יומא כמה יוסי איכא בשוקא; if not for this day, how many "Yossi"s there are in the marketplace.
And since Shavuos is an individual celebration, each individual must celebrate the Yom Tov in all its aspect, both לכם and לה'. And therefore even ר' אליעזר - who usually allows these asepcts to be divided among different people - concedes that on Shavuos each individual must celebrate both the aspect of לכם and לה' which together make up the אות, the sign of the unique covenant of Torah.