Why is an oath called a שבועה? What has it to do with the number 7? When Avraham made his oath with Avimelech, Avimelech asked "מה הנה שבע הכבשות" - What are these seven lambs? Then they made a שבועה and called the place באר שבע. It is clear that the number seven and the concept of an oath are related.
The Malbim says that in the olden days people believed that the world is governed by seven stars, so they used to take oaths in the name of these seven stars in order to prove their sincerity. [Even though Avraham is not overned by stats but he accomodated himself to them]
Another approach: There are two ways to swear and confirm that what one is saying is true - Swearing by the name of Hashem who is absolute truth [חי אני נאום השם] or swearing by the creation which consists of seven days and is also a reflection of Hashem's truth. The cycle of the world is six days and then the tachlis - the seventh day. As we say in Kiddush Levana [an apparent tautology] פועל אמת שפעולתו אמת - Hashem, the פועל [Prime Mover] is true as is פעולתו אמת, His handiwork i.e. creation which consists of seven days.
A שבועה means swearing and confirming that what I am saying is true by basing it on reality. Gehenom is the opposite of reality. It is [as the Maharal defines it] העדר - complete nothingness. Someone who takes himself out of reality must be cleansed by experiencing the non-reality that he created. Reality is אמת. The א is the source, the beginning. The מ is the middle letter which is the reality that was created. The ת, the last letter, is the tachlis. If it is not connected to the source it is מת. If it doesn't reach its purpose it is אם - if. It can only exist "if" it reaches its purpose. [A mother is called an אם which also means "if". Why?]
The holiday of שבועות is the holiday of reality, of Mattan Torah, and also the reality of the power of חכמים to create. How so? Shavuos happens ממחרת השבת after Shabbos, but not the Shabbos Hashem created [as the צדוקים claimed] but the Shabbos of the first day of Pesach which is the Shabbos that the חכמים created when they sanctified the new month and determined when the first day of Pesach will fall out. Reality is also created by תורה שבעל פה. The SEVEN weeks that are counted [a full cycle of creation] create this new reality of truth and existence. There is the week created by Hashem and then there are the weeks created by the חכמים. So we can swear either with Hashem's Name of Truth or a שבועה which is the truth of the "seven" of creation. True reality. And we celebrate שבועות which is also the time of Matan Torah - the source of reality and existence. [This explains the minhag to learn Maseches Shavuos during sefiras ha-omer]
That is the idea in a small nutshell.
Here is the expanded version.