On shabbos we get an additional soul (tannis 27b) - what is the utility of this soul? Is it the same one every week? Why would we need an extra soul?
Rashi in Maseches Beitza 16a writes that the neshama yesera is:
רוחב לב למנוחה ולשמחה ולהיות פתוח לרוחה ויאכל וישתה ואין נפשו קצה עליו
This very roughly means that it expands his heart so that he has a greater capacity for rest and joy and is able to eat and drink more without overdoing it.
Rabbi Shnuer Zalman of Liadi explains the idea of the Neshama Yesera in Torah Ohr (Parshas Vayakel pg. 87). The discourse is elucidated in the Chassidus Mevue'res series (Shabbos pg. 23), and is adapted in English here.
The basic explanation is as follows: Midrash Rabba writes that the soul is called by five names: Nefesh, Ruach, Neshama, Chaya, and Yechida. These names refer to five different levels and types of revelation within the soul of a person, with the highest level being the Yechida. This level of the soul is totally united with G-d. With the coming of Shabbos, this deepest level of a person's soul, the Yechida, is revealed within the person. This is the Additional Soul we are given on Shabbos.
It is not felt as an increase in physical life, but rather, an increase in the spiritual life force of a person: There are two types of love for G-d; one type of love is born out of contemplation of the greatness and oneness of G-d. The second is a love that is not born out of contemplation and intellect. Rather, it is a love for G-d that transcends intellect. This deep and unbounded love for G-d is the experience of our Additional Soul on Shabbos.
neshama yeterah doesn't literally mean "an extra soul", it means "extra soul" as if the amount of soul that you have is continuous. the conceptualization is for example that you have less soul when you are tired or angry and more soul when you are at rest. see footnote 5 on neshama yeterah in the tenth chapter of The Sabbath by Heschel here.