Another interpretation: "On the fifteenth day of the month." Why is the word "day" stated? To know on which day the Torah was given to Israel. Nisan, in which Israel went out from Egypt, happened to fall on the fifth day of the week; Nisan was a full month [of thirty days]; Iyyar then happened to fall on a Shabbat; Iyyar was a deficient month [of twenty-nine days]; Sivan then happened to fall on the first day of the week. And it says, "On the morrow of the Passover the children of Israel went out" (Numbers 33:3); and it says, "on the fifteenth day of the second month after they went out from the land of Egypt" (Exodus 16:1); and it says, "in the third month after the children of Israel went out" (Exodus 19:1). When you work it out, you find that you can say: in the third of the months, on the sixth of the month, on the sixth day of the week [the Torah was given].
Counting the Days from the Exodus to the Giving of the Torah
Yalkut Shimoni on Torah 257:7
דָּבָר אַחֵר, בַּחֲמִשָּׁה עָשָׂר יוֹם לַחֹדֶשׁ. לָמָּה נֶאֱמַר יוֹם, לֵידַע בְּאֵיזֶה (זְכוּת) [יוֹם] נִתְּנָה תּוֹרָה לְיִשְׂרָאֵל. נִיסָן שֶׁיָּצְאוּ בּוֹ יִשְׂרָאֵל מִמִּצְרַיִם אֵרַע לִהְיוֹת בַּחֲמִשָּׁה בְּשַׁבָּת, הִשְׁלִים נִיסָן, אֵירַע אִיָּיר לִהְיוֹת בְּשַׁבָּת, חִסֵר אִיָּיר, אֵירַע סִיוָן לִהְיוֹת בְּאֶחָד בְּשַׁבָּת, וְאוֹמֵר, "מִמָּחֳרַת הַפֶּסַח יָצְאוּ בְנֵי יִשְׂרָאֵל", וְאוֹמֵר בַּחֲמִשָּׁה עָשָׂר יוֹם לַחֹדֶשׁ הַשֵּׁנִי לְצֵאתָם מֵאֶרֶץ מִצְרָיִם, וְאוֹמֵר (שמות יט, א) "בַּחֹדֶשׁ הַשְּׁלִישִׁי לְצֵאת בְּנֵי יִשְׂרָאֵל" (כְּשֶׁתִּמְצָא לוֹמַר), [נִמְצֵאתָ אוֹמֵר] בַּשְּׁלִישִׁי לֶחֳדָשִׁים, בַּשִּׁשִּׁי בַּחֹדֶשׁ, בַּשִּׁשִּׁי בְּשַׁבָּת.