What is the source for the concept that women are on a higher spiritual level to start than men?
The Talmud [Sotah 11b] tells us about the piety of the righteous women who lived in the generation of the Exodus from Egypt. It argues that the deliverance from slavery in Egypt was a reward for that piety, and would not have happened without it. It also says that the women did not worship the Golden Calf: Only the men did, and they incurred the wrath of God.
Rabbi Isaac Luria (the Arizal) predicted in the same vein [Shaar HaGilgulim, Hakdamah 20]:
The future redemption will follow the pattern of the Exodus, and so will also come as a result of the merit of the righteous women of that generation.
R. Samson Raphael Hirsch in his commentary to Leviticus Chapter 32 asserts that they are on a higher level in at least some respect:
Thus, women's exemption from other מצוות עשה שהזמן גרמא cannot be attributed to a lower standing – as though the Torah considers women unworthy of fulfilling those mitzvos. Rather, in our view, the likely reason the Torah does not obligate women in these mitzvos is that women do not need them. For the whole purpose of מצוות עשה שהזמן גרמא is to represent – through symbolic actions – certain truths, ideas, principles, and resolutions, and to bring these values afresh to our minds, from time to time, so that we take them to heart and put them into practice. The Torah takes it for granted that woman has great fervor and faithful enthusiasm for her calling, and that the temptations awaiting her in the sphere of her calling pose but little danger to her. Hence, it was not necessary to impose on her all the mitzvos that are imposed on man. For man requires repeated exhortation to remain true to his calling, and it is necessary to repeatedly caution him against any weakness in the fulfillment of his mission. Witness מילה, which is the founding mitzvah of the Jewish people: God did not find it necessary to secure His covenant by giving women some other permanent symbol instead of מילה. Witness also the Lawgiving (Shemos 19:3): God addressed the women first, building on their fathfulness and devotion. This reality was preserved in the national consciousness and was transmitted from generation to generation. Whenever we were lost and cast down, בשכר נשים צדקניות Israel was found worthy of redemption (see Sotah 11b); it was the women who preserved and nurtured the seed of revival.
(Feldheim translation, my emphasis)
Genesis 2:17 God speaking: But of the Tree of Knowledge of good and evil you shall not eat of it, for on the day that you eat thereof, you shall surely die."
Genesis 3:3 Eve speaking: But of the fruit of the tree that is in the midst of the garden, God said, "You shall not eat of it, and you shall not touch it, lest you die.'"
It was the woman who added an extra safeguard with the additional prohibition to not touch it, whereas Gd only commanded not to eat of it
There is very little basis in Chazal to believe that women are on a higher spiritual level . There is even less basis in Chazal to believe that their higher spiritual level is the cause of their exemption from time bound mitzvahs. The earliest (equivocal) source to say such a thing is less than two hundred years old.
Conversely I know someone who, fifty years ago , when this concept was first gaining traction asked a few European born Gedolim , people who knew all of Shas and Rishonim by heart, if they knew of any source for it. Their unanimous response was "no". One of them even got very upset at hearing the concept of higher spirituality causes a women not to "need" time bound Mitzvas and felt that it is absolutely unacceptable from an Torah Hashkafa viewpoint.
There is a book "Male and Female he created them" that discusses this topic at length.