Matan Torah as the world's inner soul

איתא בגמ' דכ"ע בשבת ניתנה תורה דכתיב זכור בעצומו של יום כו'.

It is brought in the Gemara that all agree the Torah was given on Shabbos, as it is written "Remember [the Shabbos day]" — "on that very day" (Shabbos 86b).

The Gemara establishes that the giving of the Torah at Sinai took place on Shabbos, linking the words "remember" and "on that very day."

הענין הוא כמו שיש שבת ונשמה יתירה בפרט.

The matter is as follows: just as there is Shabbos and the neshamah yeseirah (extra soul) on the individual level,

Each week, Shabbos brings an "extra soul" to every individual — an added measure of inner spiritual life.

כן בכלל בריאת העולם הי' מקודם עשרה מאמרות בחי' המעשה כמ"ש ששת ימים עשה כו' בחי' נפש.

so too in the general scheme of the world's creation, first there were the Ten Utterances — the aspect of action (ma'aseh) — as it says "[for] six days Hashem made [the heavens and the earth]" (Shemos 20:11), the aspect of nefesh.

The creation through the Ten Utterances corresponds to the outer, active dimension of the world — the "nefesh," the lower vital level, formed during the six days of Creation.

אח"כ בקבלת התורה קיבל העולם הפנימיות בחי' נשמה יתירה היינו הארה מלמעלה מהטבע והוא עצמו בחי' שבת:

Afterward, at the receiving of the Torah, the world received its penimiyus (inner dimension) — the aspect of the neshamah yeseirah, namely an illumination from above nature, and this itself is the aspect of Shabbos.

Matan Torah gave the world its inner soul, an extra light from beyond the natural order — and that very inner illumination is the essence of Shabbos, which is why the Torah was given on Shabbos.

Summary: Just as Shabbos endows each person with a neshamah yeseirah, so the world as a whole was first created through the Ten Utterances as mere "nefesh" — the outer, active dimension — and only at Matan Torah, which took place on Shabbos, did it receive its penimiyus, an extra-natural illumination that is the very essence of Shabbos.